content
stringlengths 0
1.88M
| url
stringlengths 0
5.28k
|
|---|---|
The timing of the CIA's decision to cease its funding of the Free Syrian Army (FSA) is significant. With the Geneva peace talks now underway, it suggests that, unlike his predecessor Barack Obama, Trump may be serious about combating terrorism.
Throughout the conflict in Syria, which has been raging now for six years, Washington's position has lacked clarity, intelligence, or any serious moral purpose. Instead, the Obama administration went out of its way to muddy the waters, embracing the nonsensical position of combating both terrorism and those fighting terrorism at the same time. The CIA's role in funding, arming, and training the FSA, the so-called moderate rebels, only succeeded in helping to prolong the conflict and, with it, the suffering of the Syrian people - half of whom are currently displaced both internally and externally, while well over 300,000 have perished.
In the process, Obama and his supporters sought to draw a moral equivalence between a government fighting to preserve Syria as a non-sectarian, multi-religious, and multicultural secular state in which the rights of minorities and women are protected, and those intent on turning the clock back to the seventh century in the service of religious extremism, sectarianism, and obscurantism.
On the Free Syrian Army (FSA) specifically, as far back as October 2015, veteran Middle East correspondent Robert Fisk was writing that "the FSA fell to pieces, corrupted, and the 'moderates' defected all over again, this time to the Islamist Nusra Front or to ISIS [Islamic State, formerly ISIL], selling their American-supplied weapons to the highest bidder or merely retiring quietly - and wisely - to the countryside, where they maintained a few scattered checkpoints."
Even before that, in April of 2015, Erin Banco penned an article that appeared on the website of the IB Times with the self-explanatory title, 'Four Years Later, The Free Syrian Army Has Collapsed'. In her piece, she writes, "The emergence of the better-armed, ruthless Islamic State group on the battlefield in Syria last year marked the beginning of the end for the opposition groups the US dubbed the 'moderate rebels.' Now, the men and women who sparked the revolution by demonstrating in the streets of Dara'a in March of 2011 have fled, and the groups of men who took up what arms they could find to fight Assad's military and eventually became the FSA have dissipated."
The point is that the FSA long ago ceased to exist as anything other than a propaganda prop by Western governments, led by Washington, whose objective was regime change in Damascus, regardless of the fact that, if successful, such an eventuality would only have paved the way for ISIS or Nusra to assume power, with all of the ensuing catastrophic consequences involved. This is why, on this one foreign policy position alone, Obama and the Western liberal interventionist cohort he led can never be forgiven. For such people "destroying the village in order to save it" was to be Syria's fate, conforming to a pattern of destroying one country after another, while proclaiming themselves champions of democracy and human rights.
Though it remains too soon to make a considered judgment on Trump's presidency, what with the 45th President currently under siege from within Washington, and what with the mixed and incoherent messages that have emanated from his administration over the ongoing crisis in Ukraine, the status of Crimea, and belligerence when it comes to Iran, despite Tehran standing as a pillar of opposition to terrorism in the region, there are grounds for optimism over Syria. We see this also with an increase in US commitment to defeating ISIS in Iraq, where the battle to liberate Mosul is still ongoing.
While the opposition could rely on the continuing support of the US and its regional allies, it was able to adopt a rejectionist position during previous attempts to resolve the conflict and crisis at the negotiating table. Demanding that President Assad step down as a precondition of a negotiated settlement, despite being too weak on the ground to win a military victory, the opposition was able to use Washington as leverage against Russia's attempts to find a diplomatic solution.
But now, with the announcement that the CIA has decided to cease channeling funds and support to the FSA, the opposition will no longer have that option open to it, thus making the prospects of meaningful progress being made in Geneva this time round more realistic than at any other point hitherto.
That said, in Syria - where, in alliance with the so-called Syrian Democratic Forces (SDF) predominately made up of Syrian Kurds, an assault on Raqqa appears imminent - the continuing US military presence still gives cause for concern. Without the cooperation or consent of the Syrian government, US military operations in the country are a violation of Syria's sovereignty.
There is also the question of those aforementioned US regional allies to be considered. The Saudis, Qataris, and Kuwaitis have each played a malign and mendacious part in funding and supporting various Salafi-jihadist groups in Syria, which remain a potent threat, despite the recent progress that's been made by the Syrian Army and its allies on the ground. This is why serious pressure must be brought to bear by President Trump against Washington's Gulf State allies to force them to turn off the tap.
All in all, there will be much to discuss and thrash out in Geneva between all parties concerned. It promises to reveal Trump's intentions not only when it comes to fighting terrorism, but, just as crucially, when it comes to regime change. If serious about returning stability to the region, the days of Western regime change must end.
John Wight has written for newspapers and websites across the world, including the Independent, Morning Star, Huffington Post, Counterpunch, London Progressive Journal, and Foreign Policy Journal. He is also a regular commentator on RT and BBC Radio. John is currently working on a book exploring the role of the West in the Arab Spring. You can follow him on Twitter @JohnWight1
The speech by US representative Power is particularly strange to me. She gave her speech as if she was Mother Teresa herself. Please remember which country you represent. Please remember the track record of your country.
- Vitaly Ivanovich Churkin
Recent Comments
Exhibit #1 - Portugal [Link] [Link] We don't hear much in the mainstream media today about the resounding success of Portugal's decriminalization... | |
Background {#Sec1}
==========
Phytohormones are *a group of naturally occurring, organic substances which influence physiological processes at low concentrations* \[[@CR1]\]. They play crucial roles in almost all stages of plant growth and development, from embryogenesis to senescence. In addition, they also regulate response of plant to biotic and abiotic stress \[[@CR2]\]. Phytohormones have been categorized into several groups based on their structures and physiological functions, including auxins, cytokinins (CKs), abscisic acid (ABA), jasmonates (JAs), salicylates, gibberellins (GAs), ethylene (ET), brassinosteroids (BRs), polyamines, signal peptides and the more-recently-discovered hormones, strigolactones (SLs) \[[@CR1]\]. Each class of phytohormone has characteristic biological functions. However, increasing evidence shows that multiple phytohormones can mediate plant growth and development by additive, synergistic or antagonistic actions \[[@CR3]--[@CR7]\]. Phytohormone concentration and distribution are determinants of phytohormone action \[[@CR8]\]. Therefore, studies on phytohormone functions and regulation networks primarily rely on sensitive and high-throughput methods for quantification of endogenous phytohormones in plants. Accurate and simultaneous determination of multiple phytohormones enabled us to better understand the physiological functions and the regulatory networks of phytohormones. Hirano et al. \[[@CR9]\] presented the dynamic changes of each phytohormone during rice microspore/pollen (MS/POL) development by analysis of endogenous levels of ABA, CKs, GAs and IAA combined with the transcriptome results in mature anther. According to spatial and temporal distribution of CKs and the related gene function assays, Rijavec et al. \[[@CR10]\] found that CKs may perform highly contrasting roles in the filial endosperm and maternal tissues of developing seed in maize. Based on phytohormone profiling and RNA-seq analyses, Chao et al. \[[@CR11]\] discovered the specific combination of phytohormones involved in bud differentiation and shoot growth at different time points. Hence, simultaneous profiling of multiple classes of hormones, especially integrated with the results of related gene expression profilings, is a powerful tool to reveal the mechanisms and interactions of phytohormones in different growth and development stages of plants \[[@CR10]\].
There are two ways to get information about the concentrations of multiple phytohormones in plant samples. One is to divide the sample to multiple portions for independent analysis of multiple classes of phytohormones respectively \[[@CR9]\]. However, this requires a large amount of plant sample, which cannot meet the increasing demand for analysis of limited amounts of plant samples, such as a tiny organ of a rice. The other way is to develop methods for simultaneous determination of multiple phytohormones in one sample. Simultaneous analysis of multiple phytohormones is challenging due to their structural and chemical diversity, and the low contents in plant samples, usually at the nanomolar level, as well as the complex plant matrix. Therefore, it's of great significance to design a feasible strategy for simultaneous analysis of multiple phytohormones. Great efforts have been made. Additional file [1](#MOESM1){ref-type="media"}: Table S1 presents a summary of representative analytical methods for simultaneous determination of multiple phytohormones. Multiple steps involving liquid--liquid extractions or solid-phase extractions, as well as combinations of them have been used for the removal of the sample matrix and enrichment of multiple phytohormones \[[@CR12]--[@CR30]\]. Kojima et al. \[[@CR12]\] developed a multi-step strategy for determination of 43 phytohormones including auxins, ABA, GAs and CKs. The phytohormones in rice were stepwise separated into several fractions by multiple solid-phase extraction (SPE). "MS-probe" bromocholine was used for derivatization of fractions containing auxin, ABA and gibberellins to increase the MS detection sensitivity. Subsequently, phytohormones in each fraction were, respectively, analyzed using UPLC--MS/MS \[[@CR12]\]. Cao et al. \[[@CR14]\] reported a method using liquid chromatography-triple quadrupole mass spectrometry (LC--MS/MS) for the profiling and quantification of 43 phytohormones and their major metabolites, including auxins, abscisic acid, jasmonic acid, salicylic acid, cytokinins and gibberellins in a single sample extract purified by binary extraction using commercial polymer anion exchange resin (PAX) and polymer cation exchange resin (PCX), respectively. Liu et al. described a method for simultaneous analysis of 24 acidic and alkaline phytohormones, in which a binary SPE using Oasis MCX cartridges for cations and Oasis MAX cartridges for anions was employed for purification and enrichment of phytohormones. Alkaline and acidic phytohormones were eluted from different SPE cartridges, respectively. The two fractions of elution were combined for UPLC--MS/MS analysis \[[@CR15]\]. Obviously, these multiple SPE strategies were tedious and time-consuming. One-step methods can be more efficient. Recently, Meulebroek et al. \[[@CR26]\] developed a generic extraction protocol combining an UPLC-Orbitrap-MS detection method for analysis of eight phytohormones (GA3, ABA, IAA, JA, SA, Z, BA and BL) in both tomato fruit and leaf tissue. Crude plant extract was just passed through a 30 kDa Amicon^®^ Ultra centrifugal filter unit prior to LC--MS analysis. Pan et al. described a protocol for quantitative analysis of major phytohormones in crude plant extracts by high-performance liquid chromatography--mass spectrometry. Dichloromethane was used to extract and purify seven major classes phytohormones from plant extract before LC--MS analysis \[[@CR28], [@CR31]\]. Although these one-step sample preparation protocols are simple and fast, the matrix effect does exist and the efficiency of purification should be further improved. Cai et al. \[[@CR19]\] developed a method to comprehensively profile phytohormones, including 8 cytokinins (CKs), indole-3-acetic acid (IAA), abscisicacid (ABA), jasmonic acid (JA) and 10 gibberellins (GAs) by Fe~3~O~4~\@TiO~2~-based magnetic solid-phase extraction coupled with ultra-performance liquid chromatography-electrospray tandem mass spectrometry (Fe~3~O~4~\@TiO~2~-based MSPE-UPLC--MS/MS). Whereas, to date, these materials are not readily available in most laboratories.
In the current work, we have developed a clean-up strategy for profiling of multiple phytohormones, which can overcome the challenge of structural and chemical diversity. By using a one-step dispersive solid-phase extraction (DSPE) combined with UPLC--MS/MS, 54 phytohormones including auxins, ABA, SA, JA, GAs and CKs were simultaneously analyzed from a single rice sample extract. Using the developed method, we have investigated the spatiotemporal distribution of phytohormones in rice.
Results and discussion {#Sec2}
======================
Method for rapid quantification of phytohormones {#Sec3}
------------------------------------------------
Graphitized carbon black (GCB) has been widely used in QuEChERS method for sample cleanup because it can remove chlorophyll through π--π interaction with porphyrin ring of chlorophyll \[[@CR32]--[@CR34]\]. In this study, we choose GCB for cleanup of plant samples. The overall procedure for extraction and purification of phytohormones is summarized in Fig. [1](#Fig1){ref-type="fig"}. Acetonitrile is used to extract phytohormones from plant \[[@CR29]\], and graphitized carbon black (GCB) sorbent is employed for dispersive solid-phase extraction (DSPE) cleanup. In order to obtain an optimal extraction efficiency, three parameters including the amount of GCB, water content of sampling solution and extraction time were optimized (Additional file [1](#MOESM1){ref-type="media"}: Fig. S1). As the amount of GCB increased, more amount of analytes were adsorbed and less remained in the supernatant, resulting in decreased recoveries. When sampling in ACN, analytes would be adsorbed by GCB via hydrophilic interaction, so addition of H~2~O could improve the recoveries. Finally, 10 mg of GCB for DSPE, 80% ACN (v/v) for sampling and 3 min for extraction were employed for further experiments. Under the optimized conditions, the absolute recoveries of 54 phytohormones spiked in 80% ACN (v/v) and plant sample matrix were investigated respectively by using the proposed DSPE. The recoveries in standards were calculated by comparing standards that were extracted with standards without extraction. The recoveries in matrix samples were calculated by comparing samples that were spiked and then extracted with those, which were extracted and then spiked. Internal standards were added to the samples before injection to UPLC--MS/MS to calibrate errors of instrument detection. And the results are listed in Additional file [1](#MOESM1){ref-type="media"}: Table S3. Recoveries of most phytohormones in sample matrix were higher than in standards. This might be because that the sample matrix may block GCB binding sites so that less phytohormones were absorbed to the GCB and more phytohormones remained in the supernatant, resulting in higher recoveries in sample matrix.Fig. 1Schematic representation of the extraction and purification protocol for rapid quantification of phytohormones. *IS* internal standards
Ultra Performance Liquid Chromatography (UPLC) was employed for separation of 54 phytohormones. The chromatograms are shown in Fig. [2](#Fig2){ref-type="fig"}. Fifty-four tested analytes achieved baseline separation by UPLC, except for DZ7G and DZOG, MeStZ and MeScZ. Appropriate precursor to product ion transitions for each compound (54 molecular species) and their respective deuterium-labeled internal standards were determined by UPLC--ESI--MS/MS (Additional file [1](#MOESM1){ref-type="media"}: Table S2). Cytokinins and auxins were detected in the positive ion mode, ABA, JA, SA and gibberellins were identified in the negative ion mode. A polarity-switching mode enables the analysis of compounds with different preferred ionization modes. In order to enhance the sensitivity of UPLC--ESI--MS/MS, six separate functions were implemented in the MRM mode so that only ions eluted during the specified retention windows were monitored. The reproducibility and accuracy of the proposed method were evaluated with intra-day and inter-day measurements. The intra-day precisions were obtained with extractions of five samples over a day, and the inter-day precisions were obtained by extracting samples in continuous three days. The RSDs of inter- and intra-day precision were below 11.8%, and the relative recoveries were in the range of 80.3--120.4%, indicating good reproducibility and accuracy of the method (Additional file [1](#MOESM1){ref-type="media"}: Table S5). The limits of quantifications (LOQs) were calculated as the signal-to-noise ratios of 10:1 on standards with 3 replicate injections, ranging from 0.05 fmol for 2MeStZ to 29.92 fmol for cZOG in cytokinins, from 0.18 fmol for GA~1~ to 27.5 fmol for GA~9~ in gibberellins, 12.88 fmol for IAA, 93.29 fmol for SA and 1.12 fmol for JA (Additional file [1](#MOESM1){ref-type="media"}: Table S4). The LOQs are comparable with the majority of the methods in Additional file [1](#MOESM1){ref-type="media"}: Table S1. However, the presented method has the advantage of being faster than most other methods and can analyze multiple phytohormones in a single UPLC--ESI--MS/MS run (Additional file [1](#MOESM1){ref-type="media"}: Table S1).Fig. 2The MRM chromatograms of 54 phytohormones analyzed by UPLC--ESI--MS/MS. **a** Peak 1--54; **b** peak 1--7; **c** peak 22--27; **d** peak 38--42; 1, tZ7G; 2, tZ; 3, DZ; 4, cZOG; 5, DZ7G; 6, DZOG; 7, cZ; 8, DZ9G; 9, tZ9G; 10, cZ9G; 11, iP7G; 12, iP; 13, DZR; 14, tZR; 15, GA~8~; 16, cZR; 17, GA~29~; 18, iP9G; 19, 12OHJA; 20, GA~23~; 21, SA; 22, 2CltZ; 23, GA~3~; 24, iPR; 25, 2MeStZ; 26, 2MeScZ; 27, GA~1~; 28, IAA; 29, GA~6~; 30, 2MeStZR; 31, 2MeStZR; 32, ABA; 33, GA~13~; 34, GA~5~; 35, GA~19~; 36, GA~20~; 37, GA~44~; 38, JA; 39, IBA; 40, GA~34~; 41, 2MeSiP; 42, GA~51~; 43, GA~53~; 44, 2MeSiPR; 45, GA~7~; 46, GA~4~; 47, GA~24~; 48, JA-leu; 49, JA-phe; 50, GA~15~; 51, GA~9~; 52, 2BSiP; 53, GA~12~; 54, OPDA. The full names and abbreviations of the phytohormones can be found in "[Chemicals and reagents](#Sec7){ref-type="sec"}" section
Spatiotemporal distribution of phytohormones in rice {#Sec4}
----------------------------------------------------
To evaluate the spatiotemporal distribution of phytohormone species in rice. Root and leaves of rice (cv. 'Zhenshan 97B') plants were harvested at seedling stage and tillering stage. Root, senescent leaves, frag leaf and ear were harvested at grain-filling stage and mature grain stage. Then the endogenous hormone contents were analyzed. Among the 54 phytohormones investigated, 36 were quantified, including 18 CK species, 10 GA species, 5 JA species, IAA, ABA, and SA. The measurement results are shown in Additional file [1](#MOESM1){ref-type="media"}: Table S6. And the examples of chromatograms of rice tissue (rice ear at filling stage) are shown in Fig. [3](#Fig3){ref-type="fig"}. Accumulation of phytohormones displayed substantial variation in their abundance in different tissues of rice at different stages, as shown in the heat map (Fig. [4](#Fig4){ref-type="fig"}a). More species of hormones were detected in ear of rice at grain-filling stage than in other tissues, of which most showed higher concentrations. These indicated that phytohormones play important roles in seed development of rice \[[@CR35]\].Fig. 3The chromatograms of detected phytohormones in rice ear at grain-filling stage Fig. 4Spatiotemporal distribution of phytohormones in rice. **a** Heat map of spatiotemporal distribution of phytohormones. *Red* and *blue colors* indicate higher and lower concentrations, respectively. The *color scale* is shown at the *right*. Phytohormone species whose concentrations were under the quantification limit in all organs are not shown in the heat map. The value in each block is the concentration (average value, *n* = 3) as ng g^−1^ FW. *ND* not detected under the quantification limit. See Additional file [1](#MOESM1){ref-type="media"}: Table S6 for original data of measurement results. **b** Total amount of cytokinins (Total CK), cZ-type cytokinins (Total cZ-CK), cytokinin glucosides (Total gluc), and gibberellins (Total GAs) in the results of A are shown as ng g^−1^ FW. The proportions of cZ-type cytokinins \[(%) cZ-CK\] and cytokinin glucosides \[(%) Gluc\] are indicated as percentage values
In terms of cytokinins, cis-zeatin (cZ)-type cytokinins were dominant in all rice tissues investigated at all growth stages (Fig. [4](#Fig4){ref-type="fig"}b). The most abundant CK metabolite detected was cisZ-O-glucoside (cZOG) (Fig. [4](#Fig4){ref-type="fig"}a), being consistent with the previous reports \[[@CR8], [@CR36]\]. Glucosides were the major form of accumulated cytokinins in all tissues investigated, which are inactive and are thought to play a role in homeostasis of the hormones \[[@CR1]\]. To better explain the dynamic change of endogenous levels of phytohormone species in rice, a metabolic pathway was shown in Fig. [5](#Fig5){ref-type="fig"}. In root, DZR was only detected at tillering stage. Contents of iP9G increased as the plants grown up. tZ9G, cZ9G and DZ9G increased from seedling stage to filling stage, but declined at mature grain stage. In frag leaf, cytokinin precursors such as DZR and cZR accumulated in frag leaf at mature grain stage. However, glucosides decreased in frag leaf when the rice grown mature, except for tZ9G and cZOG. For ear, almost all the cytokinins investigated decreased in the mature ear, except for DZ.Fig. 5Endogenous levels of cytokinins, gibberellins and jasmonates in rice tissues at seedling stage, tillering stage, grain-filling stage and mature grain stage. The amounts of the hormones are shown as histograms with the SD (n = 3). The *y-axis* is concentration as ng g^−1^ FW. The *x-axis* represents the four growth stages. The details of each metabolic pathway are described by Hirano et al. \[[@CR9]\]
As for gibberellins, distinct tissue-specific accumulation patterns were observed. GA~7~, GA~51~ and GA~34~ were mainly accumulated in ear. Bioactive GA~4~ was detected in most of the tissues investigated except for senescent leaves at filling stage and tissues at mature grain stage. GA~7~ was only detected in ear of rice at filling stage. For root, GA precursor GA~53~ and downstream GA~51~ accumulated in root at mature grain stage. Bioactive GA~4~ showed significant reduction in root when the rice grown mature. For frag leaf, GA precursor GA~53~ and downstream GA~8~ accumulated at mature grain stage. For ear, GA precursor GA~53~ accumulated in mature ear, while the downstream GAs decreased, including the bioactive GA~4~ and GA~7~, and the deactivated GA~8~. GA~19~ accumulated in ear at grain-filing stage, but decreased to a very low level in mature ear, being consistent with the result Suzuki reported \[[@CR37]\]. The decrease in GA~19~ content at mature grain stage may indicate vigorous consumption of GA~19~, which acts as a pool GA in the biosynthetic pathway to GA~7~, GA~51~, GA~34~ and GA~8~, which highly accumulated in ear at grain-filing stage.
The rice tissues also contained large amounts of ABA, IAA, OPDA, JA and SA. ABA showed higher concentrations in leaf than in root at seedling stage and tillering stage, and the content kept steady. From grain-filling stage to mature grain stage, ABA increased in senescent leaves, but did not change in the other tissues. IAA showed the lowest accumulation in root. However, an extremely high accumulation in ear was observed. The high accumulation of IAA in ear is consistent with the highly expressed genes related to auxin biosynthetic and metabolic processes, polar auxin transport, homeostasis and auxin-mediated signaling \[[@CR35]\]. Concentration of JA declined in all tissues as the plants grown mature. Taken together, these results indicate that our analysis could show the spatiotemporal distribution pattern of the phytohormones in rice, and that the phytohormones are differentially distributed in rice tissues at different growth stages. However, for the further understanding of phytohormone function, some important clues obtained by transcriptome and other omics are needed.
Conclusions {#Sec5}
===========
In this study, we have developed a rapid one-step method for the simultaneous analysis of six groups of phytohormones, including cytokinins, auxins, salicylic acid, jasmonates, abscisic acid and gibberellins in a single run, using UPLC--ESI--MS/MS. The proposed method was successfully applied to investigate spatiotemporal distribution of multiple phytohormones in rice. The spatiotemporal information obtained may be helpful for better understanding of phytohormones functions throughout life cycle of rice when integrated into transcriptome and other omics data.
Methods {#Sec6}
=======
Chemicals and reagents {#Sec7}
----------------------
Phytohormones standards: indole-3-acetic acid (IAA), indole-3-butyric acid (IBA), abscisic acid (ABA), salicylic acid (SA), jasmonic acid (JA), 2H-jasmonic acid (2H-JA) Jasmonic acid-leucine (JA-Ieu), Jasmonic acid-phenylalanine (JA-phe), 12-OH-jasmonic acid (12-OH-JA), 12-oxophytodienoic acid (OPDA), gibberellins (GA~1~, GA~3~, GA~4~, GA~5~, GA~6~, GA~7~, GA~8~, GA~9~, GA~12~, GA~13~, GA~15~, GA~19~, GA~20~, GA~23~, GA~24~, GA~29~, GA~34~, GA~44~, GA~51~, GA~53~); trans-zeatin (tZ), cis-zeatin (cZ), transzeatin-7-glucoside (tZ7G), trans-zeatin-9-glucoside (tZ9G), cis-zeatin-9-glucoside (cZ9G), cis-zeatin-O-glucoside (cZOG), dihydrozeatin (DZ), dihydrozeatin-7-glucoside (DZ7G), dihydrozeatin-9-glucoside (DZ9G), dihydrozeatin-O-glucoside (DZOG), isopentenyladenine (iP), N^6^-isopentenyladenine-7-glucoside (iP7G), N^6^-isopentenyladenine 9-glucoside (iP9G), trans-zeatin-riboside (tZR), cis-riboside (cZR), dihydrozeatin riboside (DZR), isopentenyladenine riboside (iPR), 2-chloro-trans- zeatin (2CltZ), 2-methylthio-trans-zeatin (2MeStZ), 2-methylthio-cis-zeatin (2MeScZ), 2-methylthio-trans-zeatin-riboside (2MeStZR), 2-methylthio-cis-zeatin-riboside (2MeScZR), 2-methylthio-N^6^-isopentenyladenine (2MeSiP), 2-methylthio-N^6^- isopentenyladenine riboside (2MeSiPR), 2-benzylthio-N^6^-isopentenyladenine (2BSiP) and stable isotope-labeled standards: \[^2^H~2~\]IAA, \[^2^H~6~\]ABA, \[^2^H~4~\]SA, \[^2^H~2~\]GA~1~, \[^2^H~2~\]GA~4~, \[^2^H~2~\]GA~5~, \[^2^H~2~\]GA~6~, \[^2^H~2~\]GA~7~, \[^2^H~2~\]GA~8~, \[^2^H~2~\]GA~9~, \[^2^H~2~\]GA~12~, \[^2^H~2~\]GA~15~, \[^2^H~2~\]GA~20~, \[^2^H~2~\]GA~24~, \[^2^H~2~\]GA~34~, \[^2^H~2~\]GA~44~, \[^2^H~2~\]GA~51~, \[^2^H~2~\]GA~53~, \[^2^H~5~\]tZ, \[^15^N~4~\]cZ, \[^2^H~5~\]tZ7G, \[^2^H~5~\]tZ9G, \[^2^H~3~\]DZ, \[^2^H~5~\]DZ9G, \[^2^H~7~\]DZOG, \[^2^H~6~\]iP, \[^2^H~6~\]iP9G, \[^2^H~5~\]tZR, \[^2^H~3~\]DZR, \[^2^H~6~\]iPR were all purchased from Olchemim Ltd. (Olomouc, Czech Republic). Acetonitrile (ACN, HPLC grade) was obtained from Tedia Co. (Fairfield, OH, USA). Ultra-pure water used throughout the study was purified with Milli-Q system (Milford, MA, USA). Formic acid (FA, 88%) was purchased from Sinopharm Chemical Reagent (Shanghai, China). Graphitized carbon black (GCB) was purchased from BOSHI Biotechnology Co., Ltd (shanghai, china, <http://www.boshibio.com.cn>).
Plant materials {#Sec8}
---------------
Rice (*Oryza sativa* ssp. indica cv. Zhenshan 97B) (kindly provided by Dr. Qian Qian from State Key Laboratory of Rice Biology, China National Rice Research Institute) plants were grown under natural field conditions during the rice-growing season (from June to October). Root and leaves were harvested at seedling stage and late-tillering stage. Root, senescent leaves, frag leaf and ear were harvested at grain-filling stage and mature grain stage. All the samples were harvested at 10:00--12:00 h, placed in liquid nitrogen immediately, and stored at −80 °C. Samples were taken from three different plants per line for three biological replicates.
Sample pretreatment {#Sec9}
-------------------
As shown in Fig. [2](#Fig2){ref-type="fig"}, plant tissues (root, leaf and ear) (50 mg FW) were frozen with liquid nitrogen, grounded into powder with liquid nitrogen and then transferred into a 1.5-mL centrifuge tube. \[^2^H~2~\]IAA (1.0 ng), \[^2^H~6~\]ABA (1.0 ng), 2H-JA (1.0 ng) \[^2^H~4~\]SA (50 ng), \[^2^H~2~\]GA~1~ (0.5 ng), \[^2^H~2~\]GA~4~ (0.5 ng), \[^2^H~2~\]GA~5~ (0.5 ng), \[^2^H~2~\]GA~6~ (0.5 ng), \[^2^H~2~\]GA~7~ (0.5 ng), \[^2^H~2~\]GA~8~ (0.5 ng), \[^2^H~2~\]GA~9~ (0.5 ng), \[^2^H~2~\]GA~12~ (0.5 ng), \[^2^H~2~\]GA~15~ (0.5 ng), \[^2^H~2~\]GA~20~ (0.5 ng), \[^2^H~2~\]GA~24~ (0.5 ng), \[^2^H~2~\]GA~34~ (0.5 ng), \[^2^H~2~\]GA~44~ (0.5 ng), \[^2^H~2~\]GA~51~ (0.5 ng), \[^2^H~2~\]GA~53~ (0.5 ng), \[^2^H~5~\]tZ (0.1 ng), ^15^N~4~-cZ (0.1 ng), \[^2^H~5~\]tZ7G (0.1 ng), \[^2^H~5~\]tZ9G (0.1 ng), \[^2^H~3~\]DZ (0.1 ng), \[^2^H~5~\]DZ9G (0.1 ng), \[^2^H~7~\]DZOG (0.1 ng), \[^2^H~6~\]iP (0.1 ng), \[^2^H~6~\]iP9G (0.1 ng), \[^2^H~5~\]tZR (0.1 ng), \[^2^H~3~\]DZR (0.1 ng), \[^2^H~6~\]iPR (0.1 ng) mixture (in 5 μL ACN) was quickly added to the samples to serve as internal standards (I.S.) for the quantification. Then ACN (0.5 mL) was added and the mixture was vortexed for 30 s. After standing at −20 °C for 12 h, the supernatant was collected upon centrifugation at 10,000×*g* under 4 °C for 20 min. Subsequently, the supernatant was evaporated to dryness under a mild nitrogen stream at 35 °C and redissolved in 0.5 mL ACN containing 80% ACN (v/v). The sample solution was added into a 1.5-mLvial containing 10 mg graphitized carbon black. The mixture was vortexed vigorously for 3 min and the supernatant was transported to a 1.5-mL vial followed by evaporating to dryness. The residues were redissolved in 5% ACN (v/v) (50 μL) and 10 μL was injected for UPLC--MS/MS analysis.
Instruments and analytical conditions {#Sec10}
-------------------------------------
Analysis of phytohormones was performed on a UPLC--ESI (+/−)--MS/MS system consisting of a AB SCIEX 4500 triple quadrupole mass spectrometer (Foster City, CA, USA) with an electrospray ionization source (Turbo Ionspray), a Shimadzu LC-30AD.
HPLC system (Tokyo, Japan) with two 30AD pumps, a SIL-30AC auto sampler, a CTO-30A thermostat column compartment, and a DGU-20A5R degasser. Data acquisition and processing were performed using AB SCIEX Analyst 1.6 software (Foster City, CA, USA).
The HPLC separation was performed on a on a Shim-pack XR-ODS Ш column (75 mm × 2.0 mm i.d., 1.6 μm) purchased from Shimadzu (Tokyo, Japan) at 40 °C. A 52-min gradient of 0.1% FA (A) and ACN (B) was employed for the separation with a flow rate of 0.4 mL/min. A gradient programme of 4 min 5--5% B, 6 min 5--7% B, 10 min 7--20% B, 20 min 20--80% B, 2 min 80--5% and 5 min 5% B was used.
Multiple reaction monitoring (MRM) and the appropriate product ions were chosen to quantify phytohormones (Additional file [1](#MOESM1){ref-type="media"}: Table S2). The optimized conditions of MRM experiments were as follows: curtain gas, 40 psi; ion spray voltage, 5000 V for positive ion mode and −4500 V for negative ion mode; turbo heater temperature (TEM), 500 °C; nebulizing gas (Gas 1), 55 psi; heated gas (Gas 2), 40 psi. Data acquisition, peak integration, and the calculations were performed using Analyst 1.6.1 software (AB Sciex).
Additional file {#Sec11}
===============
**Additional file 1:** **Figure S1.** Investigation of DSPE conditions. **Table S1.** Representative analytical methods for simultaneous determination of multiple phytohormones. **Table S2.** Summary of precursor-to-product ion transitions used for the quantification of phytohormones using UPLC-ESI-MS/MS. **Table S3.** The absolute recoveries of phytohormones extracted by GCB. **Table S4.** Linearities, LODs and LOQs of 54 phytohormones by GCB-based MSPE-UPLC-MS/MS method. **Table S5.** Precisions (intra- and inter-day) and recoveries of 54 phytohormones by GCB-based MSPE-UPLC-MS/MS method. **Table S6.** Contents of detected endogenous phytohormones in rice tissues.
Wen-Jing Cai and Tian-Tian Ye contributed equally to this work
YQF designed and supervised this study. WJC and QW performed the data analysis. LY, WJC, TTY and QFZ contributed to materials collection. YQF, WJC and TTY prepared the manuscript, and all the authors critically read and approved the manuscript.
Acknowledgements {#FPar1}
================
We thank Dr. Qian Qian for providing the rice seed; Dr Yang-Sheng Li for help with the planting; Lei Yu, Quan-Fei Zhu, and Shu-Jian Zheng for their help with sample preparation and plant care. We are also grateful to the reviewers for their helpful suggestions. This work was supported by the National Natural Science Foundation of China (21475098, 91217309), and the Natural Science Foundation of Hubei Province, China (2014CFA002).
Competing interests {#FPar2}
===================
The authors declare that they have no competing interests.
Availability of data and material {#FPar3}
=================================
All data generated or analysed during this study are included in this published article \[and its supplementary information files\].
Funding {#FPar4}
=======
This work was supported by the National Natural Science Foundation of China (21475098, 91217309), and the Natural Science Foundation of Hubei Province, China (2014CFA002).
| |
27 March 2018
IAP Project Report: West Africa (2017) "Awareness Creation, Advocacy and Relevant Data Collection Strategies for the UN 2030 Agenda for Librarians"
IFLA supported an International Advocacy Programme project in West Africa in 2017. After completion of this regional project we asked the project team some questions and this is what they told us about the workshop they organised.
What were your goals?
- To build the capacity of West African librarians to effectively advocate for inclusion of libraries in development plans at different levels of governance.
- To provide skills to West African librarians for the collection and showing of their stories relating to their contributions to the UN 2030 Agenda.
How did you plan to make this happen?
We started by creating a task force confirmed by the IAP teams from Ghana and Nigeria, to submit a funding proposal to IFLA under the IAP. The project consisted in bringing together librarians from all West African countries for a two-day workshop in Accra, Ghana.
Getting experienced international librarians—preferably from West Africa—as speakers was very important to ensure a quality programme and for adopting a local approach. The Ghana Library Association hosted the event and worked together with Ghana IAP participants to arrange logistics. IAP participants from Nigeria were dedicated to preparing the content for the workshop.
Participants were hosted by the University of Ghana Guest Centre and utilised the University of Ghana Balme Library Seminar Room as the venue for the two-day workshop.
The workshop included presentations, group and individual activities and reporting, role playing, question and answer sessions, etc. Workshop materials were translated and simultaneous interpretation into English and French was offered to ensure a better understanding of the content by all participants.
How did it work?
Almost all Western African French and English speaking countries were represented at the workshop with one or two participants each, with the exception of Niger and Liberia who unfortunately could not participate.
The event was facilitated by two resource persons and supported by four IAP members from Ghana and Nigeria.
Six topics were presented covering issues such as the United Nations 2030 Agenda and the African Union's 2063 Agenda (including the role of librarians in these), Storytelling, Advocacy Cycle, Developing an Advocacy Plan and Follow-up Actions and Deadlines. There were also exercises with reporting back as a group or individually and role playing.
After the workshop, participants completed evaluation forms and received certificates of participation.
How did you use communications during the project?
In the initial stages communication was mainly handled through email and telephone. During the workshop, our face-to-face meetings were aided by simultaneous translation.
An English/French speaking IAP member from Ghana played a vital role in supporting communications between English and French speakers before, during and after the workshop.
What did you learn in the process?
The participants had many takeaways after the workshop concluded, with some key learnings:
- A deeper understanding of the UN 2030 and AU 2063 Agendas. The roles that librarians can play to support these Agendas and how to also increase the profile of the library and information profession.
- A new way of assessing our work by telling stories to each other and to the world—and more importantly to our governments and agencies.
- A clear understanding of the seven steps in the advocacy cycle and how to use it to put in place a plan to support the UN 2030 Agenda.
Role playing excercises were quite revealing and confirmed some of the issues that confront us when we meet stakeholders and helped to identify other potential issues we need to work on. We discovered there were also some issues that we had in common and others which were peculiar to individual countries; both enriched the discussions.
Few participants knew each other beforehand, but through the workshop they developed closer relationships with each other and the entire team. Communications by email and exchanges on the WhatsApp platform created at the end of the workshop were evidence of this great bond created as a byproduct of the workshop.
What are your next steps?
After the workshop, the following steps will take place:
-
Reminding and monitoring the tasks agreed by all participants, with two key activities and deadlines for 2018: | http://origin-www.ifla.org/node/36070 |
Supporting our community to create sustainable and positive change in the health of our community.
At Augusta Health, we are dedicated to supporting community nonprofit agencies that help to promote the health and well-being of our community.
Our Community Contributions funding directly supports programs and initiatives that seek to improve the health of our community in one of the areas of need identified in the Community Health Needs Assessment (CHNA).
We welcome requests for Community Contributions funding that meet the following guidelines:
- Applicant is a community nonprofit agency or local government in Augusta County, City of Staunton and City of Waynesboro. Requests from Highland County, Rockbridge County and Bath County may also be considered. Individuals and for-profit businesses are not eligible.
- Funds will be used for costs directly related to the program or event described in the application.
- Funds will be used for costs incurred in the year in which the funds are received.
The maximum amount available for a recipient is $2,500. Applications are considered monthly throughout the year.
Please note that Augusta Health will not consider requests that are for the sole benefit of an individual, a political organization, or organizations/programs in potential conflict with the mission, vision and values of our organization.
Apply now for a Community Contribution from Augusta Health. | https://www.augustahealth.com/service/community-outreach/strategic-funding/community-contributions/ |
How long does it take to roast a turkey?
How long to cook a turkeyWeight of Bird Roasting Time (Unstuffed) Roasting Time (Stuffed)10 to 18 pounds3 to 3-1/2 hours3-3/4 to 4-1/2 hours18 to 22 pounds3-1/2 to 4 hours4-1/2 to 5 hours22 to 24 pounds4 to 4-1/2 hours5 to 5-1/2 hours24 to 29 pounds4-1/2 to 5 hours5-1/2 to 6-1/4 hours28 мая 2014 г.
Do you roast a turkey covered or not?
Q: Should I roast the bird covered or uncovered? A: The Butterball folks recommend cooking the turkey uncovered in a roasting pan. Two-thirds of the way through cooking, Butterball says foil can be placed over the breast area to prevent it from drying out.
How do I keep my turkey moist?
For moist meat without the hassle of clearing fridge space to soak the bird in a vat of brining liquid, try a dry brine. Salting a turkey and letting it rest before roasting seasons it deeply and helps it retain moisture.
Do you put water in bottom of roasting pan when cooking a turkey?
No. We do not recommend adding water to the pan because it creates a steam and may steam-burn the turkey. The turkey will produce its own flavorful juices. After cooking, you can extend the turkey’s juices with broth or wine, then add it to your gravy for extra flavor.
What is the best temperature to cook a turkey?
What temperature to cook the turkey? Preheat the oven to 450°F then drop the temperature to 350°F after putting the turkey into the oven. What temperature should the turkey be? The turkey is done when it registers a minimum of 165° in the thickest part of the thigh.
Should I start my turkey at a higher temperature?
I find it is best to start the turkey at a fairly high temperature (400°F), roast for about twenty minutes and then lower the heat to 350°F for the remainder of the cooking time. Sometimes I forget to lower the oven, though, and the turkey still comes out fine, just perhaps a little darker than I would like!
Do you cook a turkey at 325 or 350?
Roast the turkey uncovered at a temperature ranging from 325°F to 350°F. Higher temperatures may cause the meat to dry out, but this is preferable to temperatures that are too low which may not allow the interior of the turkey to cook to a safe temperature.
Does stuffing a turkey make it more moist?
Our Foolproof Tips for Stuffing a Turkey. Cooking your stuffing inside the Thanksgiving turkey gives it an unparalleled flavor and texture. As the bird roasts, its juices are absorbed into the stuffing, resulting in a savory, moist, delicious mixture that’s hard to achieve any other way.
Why is my turkey always dry?
Because dark meat has more connective tissues, it takes longer to break down, so if you cook the turkey whole, by time the legs and thighs are done, the breasts are overcooked and dry. … After cooking, let the meat rest until it’s close to room temperature in order to let the juices redistribute.
Should I put butter or oil on my turkey?
Don’t butter your bird
Placing butter under the skin won’t make the meat juicier, though it might help the skin brown faster. However, butter is about 17 percent water, and it will make your bird splotchy, says López-Alt. Instead, rub the skin with vegetable oil before you roast.
Can I season my turkey the night before?
The night before your turkey goes in the oven, season it with a blend. Maybe fresh herbs, garlic, shallots and/or onions, with some olive or whatever your tradition. … Before roasting, remove the turkey from the brine, rinse well under running water, and pat dry inside and out with paper towels. | https://parwarestaurante.com/turkey-cooking-tips/cooking-a-turkey-roast.html |
Fiona loves sharing the grace and precision of Iyengar Yoga. These classes help develop strength and confidence whilst creating a sense of lightness and harmony throughout the body and mind.
Classes are structured to allow for progression, so booking for the calendar month is encouraged, especially for new students. For details contact Fiona - www.flyoga.com.au
Please log in to view your upcoming appointments. | https://divinewellbeingcentre.com.au/trainer/fiona-lisle-classes/ |
Mediation is the attempt to help parties in a disagreement to hear one another, to minimize the harm that can come from continuing the disagreement and to maximize the opportunity to reach an agreement. The goal is to find a way of preventing the areas of disagreement from interfering with the process of seeking a compromise or mutually agreed upon outcome.
Maureen Binetti has been practicing law for over 35 years and often serves as an independent investigator of internal employee complaints, and as a state-approved mediator of employment claims. She has investigated all types of internal workplace complaints over 25 years, often resulting in the amicable resolution of potential legal issues. In addition, Ms. Binetti has successfully mediated more than 550 employment law claims over the past 15 years, many pursuant to the early mediation requirement of all employment disputes by the New Jersey state courts, thereby helping parties avoid costly, risky and emotionally exhausting trials by resolving the majority of the disputes at an early stage, prior to discovery.
Effective and knowledgeable Employment Law mediators are relatively rare. Securing a mediator with deep experience and knowledge of employment law will stack the odds of resolving an employment-related dispute in your favor. Seasoned employment law mediators are adept at identifying the often complex and nuanced issues that often define employment disputes. Generally, emotions of the parties involved may be strong and oftentimes solutions are the result of creative thinking and the ability to articulate the underlying issues. Optimally, employment disputes in mediation are best served by retaining an employment law mediator who has experience handling cases in state and federal court on behalf of both employers and employees. A mediator must persuade the parties to consider the risks and benefits of continuing the dispute versus settlement. A mediator that handles cases on both sides has the ability to identify and articulate the strengths and weaknesses of both sides, and the perspective to predict the likelihood of claims should the litigation continue. | https://www.wilentz.com/personal/employee-rights-and-responsibilities/mediation |
This is a read note of Mastering Ethereum Ch11: Oracles. Oracles are systems that can provide external data sources to Ethereum smart contracts. Ideally oracles are systems that are trustless, meaning that they do not need to be trusted because they operate on decentralized principles.
1 Why Oracles Are Needed
In order to maintain consensus, EVM execution must be totally deterministic and based only on the shared context of the Ethereum state and signed transactions. This has two particularly important consequences: the first is that there can be no intrinsic source of randomness for the EVM and smart contracts to work with; the second is that extrinsic data can only be introduced as the data payload of a transaction.
Oracles, ideally, provide a trustless (or at least near-trustless) way of getting extrinsic (i.e., “real-world” or off-chain) information, such as the results of football games, the price of gold, or truly random numbers, onto the Ethereum platform for smart contracts to use. They can also be used to relay data securely to DApp frontends directly. Oracles can therefore be thought of as a mechanism for bridging the gap between the off-chain world and smart contracts. However, this can also introduce external risks to Ethereum’s security model because anyone can hack the oracle.
Note that some oracles provide data that is particular to a specific private data source, such as academic certificates or government IDs. As such, these data sources count as “oracles” because they also provide a data bridge for smart contracts. The data they provide generally takes the form of attestations, such as passports or records of achievement. Attestations will become a big part of the success of blockchain platforms in the future, particularly in relation to the related issues of verifying identity or reputation, so it is important to explore how they can be served by blockchain platforms.
Oracles can also be used to perform arbitrary computation, a function that can be especially useful given Ethereum’s inherent block gas limit and comparatively expensive computation costs.
2 Oracle Design Patterns
The key functions of an Oracle are:
- Collect data from an off-chain source.
- Transfer the data on-chain with a signed message.
- Make the data available by putting it in a smart contract’s storage.
Once the data is available in a smart contract’s storage, it can be accessed by other smart contracts via message calls that invoke a “retrieve” function of the oracle’s smart contract; it can also be accessed by Ethereum nodes or network-enabled clients directly by “looking into” the oracle’s storage.
The three main ways to set up an oracle can be categorized as request–response, publish-subscribe, and immediate-read.
- immediate-read oracles are those that provide data that is only needed for an immediate decision
- publish–subscribe oracles provide a broadcast service for data that is expected to change (perhaps both regularly and frequently) is either polled by a smart contract on-chain, or watched by an off-chain daemon for updates.
- request–response oracle might be implemented as a system of on-chain smart contracts and off-chain infrastructure used to monitor requests and retrieve and return data.
3 Data Authentication
There is a distinct possibility that data may be tampered with in transit, so it is critical that off-chain methods are able to attest to the returned data’s integrity. Two common approaches to data authentication are authenticity proofs and trusted execution environments (TEEs).
Authenticity proofs are cryptographic guarantees that data has not been tampered with. Based on a variety of attestation techniques (e.g., digitally signed proofs), they effectively shift the trust from the data carrier to the attestor (i.e., the provider of the attestation). By verifying the authenticity proof on-chain, smart contracts are able to verify the integrity of the data before operating upon it.
TEE methods utilize hardware-based secure enclaves to ensure data integrity.
4 Decentralized Oracles
ChainLink has proposed a decentralized oracle network consisting of three key smart contracts (a reputation contract, an order-matching contract, and an aggregation contract) and an off-chain registry of data providers. The reputation contract is used to keep track of data providers' performance. Scores in the reputation contract are used to populate the off-chain registry. The order-matching contract selects bids from oracles using the reputation contract. It then finalizes a service-level agreement, which includes query parameters and the number of oracles required. This means that the purchaser needn’t transact with the individual oracles directly. The aggregation contract collects responses (submitted using a commit–reveal scheme) from multiple oracles, calculates the final collective result of the query, and finally feeds the results back into the reputation contract. | https://www.mindissoftware.com/post/2021/12/mastering-eth-11/ |
Over the past 15 years the number of people suffering from chronic hunger has dropped across the globe – thanks at least in part to the implementation of the Millennium Development Goals that were agreed in 2000. Nevertheless, there are still nearly 800 million people who are suffering from hunger. A total of about two billion people are suffering from micronutrient deficiencies. Infants in particular suffer damage to their health as a result of hunger and malnourishment, damage that cannot be reversed when they are older.
And yet, a world without hunger is possible! In a statement formulated as a result of an initiative launched by the German Development Ministry BMZ, the G7 announced at Schloss Elmau that, working in cooperation with their partners, they aim to lift 500 million people out of hunger and malnutrition by 2030. The G7 are thus sending a strong signal to the international community that we are living up to our responsibility as the world's strongest economies, not least vis-à-vis poorer countries. Our intention is also to contribute significantly to the Sustainable Development Goals that are to be adopted at the special UN summit in September. Hunger and malnutrition need to be addressed if people are to have any chance of living decent lives, and in order to give them the possibility to play an active part in their societies and economies.
We are making a point of focusing on people and thus on the impact we want to achieve. The means for achieving this impact will be increased efficiency, even better coordination among ourselves and with our partners, more targeted action, and using our financial resources in the right way.
In a separate annex to the Summit Communiqué, a broad Food Security and Nutrition Development Approach was adopted. The Approach outlines areas which are important for achieving th above-mentioned goal: focusing on an inclusive rural development approach, fostering responsible investment and sustainable agriculture, providing targeted assistance for malnourished children and women – one of the most effective ways of boosting empowered development – and improving our support for hungry people in conflict and crisis settings. The Approach was drawn up by the German Development Ministry BMZ and the other G7 partners working together in the G7 Food Security Working Group. It highlights the need to address many different areas in order to attain food and nutrition security. This goes beyond increasing agricultural production and encouraging investment in agriculture.
Through its special initiative ONE WORLD – No Hunger, the German Development Ministry BMZ has already significantly expanded its assistance in the fields of food security, rural development and agriculture. The aim of the special initiative is, firstly, to stop hunger and malnutrition among the current global population and, secondly, to make sure that future generations in a growing world population will also be able to feed themselves.
People currently suffering from starvation and malnutrition, especially infants and pregnant and nursing women, must have access to adequate, affordable and healthy food supplies, health care services and safe water. We intend to provide encouragement and support to our partner countries and help them implement the right to food. No progress can be made to address these issues in the absence of sustainable natural resource protection and management. (...) Resource use and its associated impacts are very unevenly distributed across different countries and demographic groups. The survival of poor and marginalised groups in rural areas depends crucially on access to arable land, pristine forests and clean water.
from the Charter for the Future "ONE WORLD – Our Responsibility" | https://www.bmz.de/g7/en/Entwicklungspolitische_Schwerpunkte/Ernaehrung_sichern/index.html |
Blackrock House is on the Causeway Coastal Route, one of the worldâs greatest coastal drives, where you journey along the winding seaside, dramatic white chalk cliffs and emerald glens all washed by the bracing Wild Atlantic Ocean. We are a 15 min drive to Northern Irelandâs famous UNESCO World Heritage Site, the Giant's Causeway.
Step on thousands of basalt columns tumbling down into the Atlantic Ocean on the grand causeway. Itâs an epic sight, with a whopping 40,000 hexagonal-shaped steppingstones, which date back to a volcanic age almost 60 million years ago. This iconic, geological landscape also tells stories of myths and legends where Irish Giants battles Scottish rivals and Spanish sailors shipwrecked with their golden treasure.
There are numerous paths and walkways, the most spectacular is the five-mile clifftop walk along this stunning stretch of the Causeway coastline which gets you away from the crowds. Pack you walking shoes and we will pack your picnic! Guided personal tours can be arranged with local guides and pick ups from Blackrock. | https://www.blackrockbandbportrush.com/giants-causeway-accommodation |
Using telescopes of the European Southern Observatory (ESO), astronomers discovered giant magnetic spots on the surface of very hot stars in star clusters. In addition to spots, some of these stars also have super-powerful flashes, explosive events, in which several million times more energy is released than with similar phenomena on the Sun. The study was published in the journal Nature Astronomy and is also reported in the ESO press release.
Scientists led by Yazan Momany of the Astronomical Observatory in Padua (INAF, Italy) investigated a special type of stars called stars of the extreme horizontal branch. These are objects with masses about half that of the Sun, but 4–5 times hotter than it. “These hot and small stars are distinguished by the fact that, as we know, they bypass one of the final stages of a typical star’s life and die prematurely,” Momani says. “In our Galaxy, these peculiar, unusual hot objects are usually part of a close binary system.”
However, it unexpectedly turned out that in the vast majority of stars of the extreme horizontal branch observed in densely populated star groups – globular clusters, no companion stars were noticeable. In addition, during long-term monitoring of such stars with ESO telescopes, the group discovered a new feature in these mysterious objects. Observing three globular clusters, Momani and his colleagues found regular changes in brightness in many stars of the extreme horizontal branch contained in them with periods of only a few days to several weeks. “After all the other scenarios were eliminated, there was only one opportunity to explain the observed brightness variability,” said Simone Zaggia, a researcher at the Astronomical Observatory in Padua (INAF, Italy) and a former ESO scholarship holder. “These stars must be stained!”
The spots on the stars of the extreme horizontal branch turn out to be completely different from the dark spots on our Sun, although in both cases they are caused by magnetic fields. The spots on these hot and extreme stars are brighter and hotter than the surrounding surface of the star, in contrast to the Sun, where the spots look like dark marks against a background of brighter and hotter surroundings. In addition, the spots on the stars of the extreme horizontal branch are much larger than the sun – they cover up to a quarter of the surface of the star. These spots are incredibly stable – they last for decades, and individual sunspots are short-lived formations, they last from several days to several months. When hot stars rotate, spots on the surface either appear in the field of view or disappear from it, which leads to the observed changes in brightness.
In addition to the brightness variability associated with the spots, the group discovered a pair of stars of the extreme horizontal branch on which superflashes occurred – explosive release of energy, which also indicates the presence of a magnetic field. “They look like flashes that we see on the sun, but are ten million times more powerful,” says Henry Boffin, an astronomer at ESO headquarters in Germany. – We did not expect such behavior in these stars; it shows how important it is to take magnetic fields into account when explaining their properties. ”
After sixty years of trying to understand the nature of the stars of the extreme horizontal branch, astronomers now have a more complete picture of their behavior. Moreover, this discovery can help explain the origin of strong magnetic fields in many white dwarfs – objects representing the last stage of the life of sun-like stars and having much in common with the stars of the extreme horizontal branch. “The main thing,” says team member David Jones, a former ESO scholar, and now an employee of the Canary Islands Institute of Astrophysics in Spain, “that the brightness changes of all the hot stars, from young stars like the Sun to old stars of the extreme horizontal branch and long dead white dwarfs can now be connected to each other. All these objects can have magnetic spots on the surface. ”
To reach this conclusion, astronomers used several ESO Very Large Telescope (VLT) receivers, including VIMOS, FLAMES and FORS2, as well as an OmegaCAM camera mounted on the VLT Survey Telescope at the Paranal Observatory. They also used the ULTRACAM receiver on the New Technology Telescope at the ESO La Silla Observatory, also in Chile. The breakthrough occurred when the group conducted observations of stars in the near ultraviolet region of the spectrum, which allowed astronomers to identify extremely hot stars among colder in globular clusters. | https://usnewslatest.com/astronomers-have-discovered-giant-magnetic-spots-on-hot-stars/?amp=1 |
Jamaica’s Usain Bolt clocked a 100m world record of 9,72 seconds on Saturday to electrify the Reebok Grand Prix athletics meeting.
The 21-year-old broke the previous record of 9,74 set by compatriot Asafa Powell in Rieti, Italy, on September 9 2007.
With a favourable wind of 1,7m/sec, Bolt finished ahead of 100m and 200m World Champion Tyson Gay of the United States (9,85) and American Darvis Patton (10,07).
On a night when thunderstorms and the threat of lightning forced a 45-minute disruption to the action — and that after the start of the meet was delayed for an hour — Bolt delivered the real jolt of the night.
The 1,95m-tall Jamaican immediately became the man to beat as the athletics season builds toward the Beijing Olympics in August, with Gay, Powell and the rest of the world’s sprinters relegated to the role of challengers.
”This world record doesn’t mean a thing unless I get the Olympic gold medal, or win at the World Championships,” he said.
Bolt, the 200m world championships silver medallist, had set the athletics world buzzing on May 3 when he clocked 9,76 — then the second-fastest time in history — at a meeting in Kingston.
With that performance he appeared poised to live up to his earlier credentials, which included world junior records and status as the youngest man to reach a World Championship sprint final, at Helsinki in 2005.
While Bolt is now front and centre in the 100m reckoning, he said the 200m remains his passion.
”I always say the 200 is my favourite race. That’s not going to change,” said Bolt, who is considered by many a likely threat to Michael Johnson’s 200m world record of 19,32 set in Atlanta in 1996.
On the same East River island in New York City — but at a different stadium — that saw Leroy Burrell and Frank Budd set previous 100m world records, Bolt blazed out of the blocks and was never threatened.
”I knew if I got out of the blocks OK, I’d have a good chance,” Bolt said. ”I knew this was a fast track and that I was ready to run something in the 9,7’s.
”But 9,72, that’s pretty good. When I saw the time they put on the board [at first 9,71] I realised it was something special.”
Gay applauded the performance, but insisted it didn’t change his approach to Beijing.
”I was only 1/100th off my own PR [personal record], so you can see what a great race Usain ran tonight [Saturday],” Gay said. | https://mg.co.za/article/2008-06-01-bolt-electrifies-with-new-100m-world-record/ |
# Phonogram (linguistics)
A phonogram is a grapheme (written character) which represents a phoneme (speech sound) or combination of phonemes, such as the letters of the Latin alphabet or Korean letter Hangul. For example, "igh" is an English-language phonogram that represents the /aɪ/ sound in "high". Whereas the word phonemes refers to the sounds, the word phonogram refers to the letter(s) that represent that sound.
Phonograms contrast with logograms, which represent words and morphemes (meaningful units of language), and determinatives, silent characters used to mark semantic categories. | https://en.wikipedia.org/wiki/Phonogram_(linguistics) |
The effect of dose on maternal-foetal transfer of fluoride in rabbits.
Placental transfer of fluoride was investigated by fluoride determination in the bones and teeth of newborn rabbits whose mothers had been treated with fluoride during pregnancy. The mothers were given doses of 0, 0.10, 0.52 and 1.05 mmol fluoride per kg body weight as sodium fluoride, from the 16th day after conception to the end of pregnancy. All the doses produced a significant increase of fluoride level in the bones and teeth of newborn rabbits, indicating that the placenta was no barrier for the passage of fluoride.
21473324 - Obesity in pregnancy: addressing the issues at the booking appointment.
701534 - Subcutaneous fat necrosis of the newborn.
15732064 - Place of preimplantation diagnosis in genetic practice.
Department of Toxicological Chemistry, School of Pharmacy, University of Belgrade, Yugoslavia. | http://www.biomedsearch.com/nih/effect-dose-maternal-foetal-transfer/1854264.html |
The Careers Department is responsible for the organisation and arranging of various activities in school.
It is a statutory requirement for maintained schools and academies to provide Years 8-11 with the opportunity to access information regarding technical education qualifications and apprenticeships.
Students will be given the opportunity to gain an insight into the work of professionals and apprentices from industries and providers during a range of initiatives including the careers interview programme and a wide range of school careers events which include:
- Careers Thursdays, where outside speakers talk about their career path, a particular profession and apprenticeship opportunities.
- Careers and progression Interviews
- The Morrisby test in Year 10 (psychometric testing) to support career choices
- Higher Education Evening in Year 12
- Sixth Form practice interviews which take place at the end of Year 12 to help prepare students for forthcoming university, apprenticeship or job interviews.
- We are a partner school with InvestIN Education
- Students in Years 10 to 13 have access to the Springpod platform to find work experience and careers ambassadors.
- Careers Information is also delivered during PSHE.
Our objective is to deliver independent advice to our students so that they can make constructive choices to achieve their aims and ultimately fulfil their ambitions. This advice is available throughout the year via the online resources in this section, through the careers areas located in both school libraries and through independent advice at certain points of the academic year.
All years are encouraged to visit the Careers Library, especially when choosing GCSE, A-Level options, and Higher Education Courses. Our Careers Policy Statement is available to download below and via the Policies page of this website.
Library Resources
The School has two libraries and each contains careers resources relevant to students.
The Main School Library has a selection of books and magazines which are regularly updated, containing information and advice on choices of subjects towards future routes into career choices.
The Sixth Form Library has two areas:
The Careers section containing information on how to get into various careers, student finance, scholarships and bursaries, working abroad, voluntary organisations, and much more.
The University section contains current prospectuses from a wide range of Higher Education institutions, together with information on Open Days, Courses and Gap Year opportunities.
Online Resources
There are numerous websites which can offer a range of information in relation to careers and higher education. Please see our Online Resources page here.
The government website for careers guidance is the National Careers Service.
Further information relating to the St. Olave's Grammar School Careers Provision is available here. | https://www.saintolaves.net/38/careers |
This week, the Lake Wales Public Library encourages all community members to visit their website to explore and access virtual services and programs.
The Lake Wales Public Library offers a wide array of online resources that are available from the comfort of home, including online training resources through Gale Courses and Gale Presents Udemy. The website also features downloadable eBooks, audiobooks, movies, television and magazines with their OverDrive, Hoopla and Flipster databases.
Don't miss wonderful educational and research tools through the Florida Electronic Library, genealogy research via HeritageQuest and Ancestry, as well as Lake Wales history archives including digitized newspapers, photos, maps and documents.
The Lake Wales Library Association and Florida's Natural Growers Foundation are sponsoring a National Library Week contest, Library App Digital Scavenger Hunt. The contest will enhance library users' knowledge of the Polk County Library Cooperative's App. Download the app, use the scavenger hunt form to answer eleven questions, turn the form in via email or at the Lake Wales Public Library. Request a copy of the Scavenger Hunt by emailing [email protected], or call 863-678-4004, extension 0. Entries must be received by 2 p.m. Saturday, April 10. A drawing will be held for a $50 gift card, as well as several Vera Bradley paperback book covers.
April 4-10, 2021 is National Library Week, a time to highlight the essential role libraries, librarians and library workers play in transforming lives and strengthening communities. The theme for this year's National Library Week is Welcome to your library, which promotes the idea that libraries extend far beyond the four walls of a building and that everyone is welcome to use their services.
During these challenging times, libraries of all types have been going above and beyond to adapt to our changing world by expanding their resources and continuing to meet the needs of their patrons. Libraries across the country are making a difference in people's lives by providing electronic learning resources like online homework help and free wi-fi access for students and workers who may lack internet access at home.
First sponsored in 1958, National Library Week is a national observance sponsored by the American Library Association and libraries of all types across the country each April.
For more information, visit the library's website at http://www.lakewalesfl.gov/library, or call 863.678.4004 ext. 0. | https://www.lakewalesnews.net/story/2021/04/07/ridge-life/scavenger-hunt-helps-celebrate-national-library-week/2193.html |
Missing in Connecticut: Investigators hope circulating photos will help identify unknown remains
Some investigators with long unsolved cases of unidentified bodies have decided to post photographs of them online to try to identify them.
The Milwaukee County medical examiner's office in Wisconsin began putting photographs of unidentified deceased on its website in December, and the site now features cases going back to the 1970s.
Michael Simley, a forensic investigator with the Milwaukee office, said its new website feature so far hasn't led to any positive identification of remains, but it has led to tips and promising leads. According to Simley, it is modeled after similar programs in other jurisdictions that have had success identifying people this way.
"I think it is a fantastic tool for when leads have gone cold," Simley said. "It takes a lot of hard work to identify these people. It is another resource. For the families of missing people, the not knowing is a problem. They just want closure. We are not doing this just to show the pictures, but to bring closure to families and bring names back to these people."
RELATED: Giving a name to the 'missing missing'
FACEBOOK PAGE: Missing in CT
The site includes warnings that the images may be disturbing to some people. Among the cases featured on the website are a woman found floating in the Milwaukee River in 1983, and a woman floating in Lake Michigan in 1974.
It includes links to the U.S. Department of Justice's National Missing and Unidentified Persons System, or NamUs, where the images are also posted.
Most entries on NamUs for the unidentified dead don't include an actual photo of the deceased, and no photographs are on NamUs for the 41 cases of unidentified dead found in Connecticut.
Police in Greater New Haven had mixed reactions to whether photographs of the state's unidentified dead should be put online. They noted that such a program would only be helpful in cases where the person is found shortly after death and is still recognizable.
As for East Haven's August 1975 case of a still unidentified female murder victim, police Detective Sgt. Bruce Scobie said he doesn't believe anyone could identify her from photographs.
"In this case, I don't believe it would accomplish anything," Scobie said. "It may help with other cases. You'd have to weigh what you might get out of it versus how a family might feel to see a loved one dead on the Internet."
East Haven has circulated an artist's rendering of the woman.
Janice Smolinski of Cheshire, whose son William Smolinski Jr. is missing, said she hopes more investigators will put photographs of deceased online, as she would look occasionally and check for her son.
"I feel all medical examiners should release the photographs of unidentified deceased," Janice Smolinski said. "Families go through a tremendous amount of heartache and pain knowing a loved one is out there somewhere. By posting the deceased individual for others to see, matches will be possible. This means long awaited resolution for a loved one."
State police spokesman Lt. J. Paul Vance said after viewing the Milwaukee website, "Sometimes you have to go to extremes to solve a case."
Call Michelle Tuccitto Sullo at 203-789-5707. | https://www.nhregister.com/news/article/Missing-in-Connecticut-Investigators-hope-11547262.php |
The BA (Hons) Music Performance is focused on Western art music, with training in practical musicianship, music theory, and academic knowledge. It is a practice-based, creative degree programme in which you would be expected to practice individually, rehearse in ensembles, and perform in both contexts. During your studies, you will also gain knowledge in music theory, music history, composition, arrangement, music technology and music education.
This programme is ideal for you if you have a background in performance with an interest in solo performance and/or ensemble playing. Graduates will be equipped for a professional career in the music industry as performers, educators, and session musicians, utilising a broad exposure to other related areas, including composition and arrangement. Furthermore, due to the interdisciplinary nature of the programme and its strong emphasis on collaborative work, you would gain a wide range of transferable skills necessary for employment in today’s creative industries.
STPM
A-Level
Australian Matriculation
Canadian Matriculation
Monash University Foundation Year
Sunway Foundation in Arts
Sunway Foundation in Science Technology
Unified Examination Certificate
International Baccalaureate
Sunway Diploma in Related Field*
Specific Requirements
English Language Requirements
YEAR 1
YEAR 2
YEAR 3
Copyrights © 2023-24 Education Malaysia. All rights reserved. | https://www.educationmalaysia.in/university/sunway-university/course/ba-hons-music-performance |
Written By Keshav Mohta (Grade 8)
The South China Sea (a marginal sea that is part of the Pacific Ocean) has been a disputed area for very long. It is the sea stretching from the Malacca Straits to Taiwan, neighbouring the coasts of eight countries: China, Taiwan, the Philippines, Malaysia, Brunei, Indonesia, Singapore, and Vietnam.
The sea has two major sets of uninhabited islands. The sea is important to the nations around for multiple reasons:
- Oil and Gas resources
- Fishing Opportunities
- A busy shipping route
The issue dates all the way back to 1947, where the Chinese government put in place the eleven-dotted line claiming almost the whole of the sea. Later China under the new government published a map in 1953 where only 9 of 11 lines remained. The situation was peaceful, till the past 2 decades, when China has started militarising the sea and started setting up naval bases on the islands.
The major players in the game are China, Vietnam, Philippines, Taiwan and the US. The first four of these have multiple claims on different territories of the sea. China, Vietnam and Taiwan essentially claim the region on the basis of historical account. Philippines claims a part of it under a UN convention which allows it to have 200 nautical miles of the sea from its coast under its complete authority.
Currently the sea is getting militarised day after day and incidents between competing countries are becoming increasingly frequent, elevating the risk of escalation. The issue also draws interest to many other countries who are directly or indirectly involved. The US, Japan and India also have a lot at stake.
The US doesn’t want the sea going into the hands of the Chinese who could monopolize trade given the sea’s geography. India has deals with Vietnam for oil mining in the sea and Japan is at a risk of Chinese occupation in the North China Sea. The Philippines have filed a case at the Hague (International Court of Justice), in light of the recent Chinese military aggression. The other nations involved are only a few inches away from what we can call a Mexican stand-off.
The situation is only escalating as the US, runs military campaigns across the sea with its allies, Philippines and Vietnam. Other stakeholders are not too far behind and the situation could certainly escalate into the dreaded World War III. As Nathan Bedford Forrest (a confederate general during the American civil war) once famously said – “War means fighting, and fighting means killing.”
This article was originally published in I Kid You Not. | https://mypenmyfriend.com/the-south-china-sea-dispute-and-how-it-will-impact-the-world/ |
It was a cool summer’s evening while watching Dustin Brown’s victory over Rafael Nadal during their recent tennis match…but the victory was bigger than a qualifying round of tennis. The real victory was that I was able to watch this tennis match within a social environment that understood shared values and mentality are the things that really unite people rather than the human illusion of race.
You could argue that is because I just so happen live in London, which just so happens to be one of the greatest, multi-cultural, over crowed, overpriced cities in the world!
I can safely say that I was in the company of a collective that saw this tennis match as a victory for Germany over Spain without ever bringing the hue of Dustin Brown’s skin into question. I think large modern sporting events have a great track record for seeing individual athletes as representative of a country no matter what they look like.
It’s safe to say that over the years evolution and the external environment have a lot to do with physical traits and characteristics that different people have developed…but that doesn’t change the fact that if a man from Japan engages in a sexual union with a woman from Norway the only thing that will be produced is another human! That new human will not be “bi-racial” or of “mixed race” but rather a child of the human race who has a heritage of two different cultures.
Fortunately a collection of scientific minds at UNESCO (United Nations Educational, Scientific and Cultural Organization) have also reached the same conclusion through their years of methodical study. Anthropologists have known for a while now that there are no major sets of behaviours that are directly connected to a “race” but rather certain ways of being are connected to groups of people who have a similar commonality of experience.
Now there is nothing wrong with having a physical trait that you find attractive…I’ve always been a sucker for great teeth and dazzling smile myself…but the more time moves us forward the more we’ll realise the social illusion of race is something people have used to serve their own social agenda to oppress certain groups.
As much as I do have a secret love affair with American culture, the historical dysfunction and amazing progress of the American experience has highlighted the importance of diversity but corrupted the idea of race. Of course there are always going to be people who foolishly make judgements based the melanin content (or lack thereof) in someone’s skin and I am in no way trying to belittle or dismiss the experience of someone who has been unfairly judged along colour lines.
We know for certain there are different cultures, music, food, fashions, languages, dance, arts, philosophies, colours, customs and ethnicities depending on circumstances you just so happened to be born into…but none of that changes the fact that we are all the same damn species. The rules of the game have been changing for a while now and what unites people is not necessarily the melanin content in an individuals skin, but a shared cultural identity and mentality… just ask the people attending the San Diego comic con this year!
Until next time! | https://tyronepierre.com/2015/07/08/we-are-all-in-the-human-race/ |
“Mend the Gap : Smart Integration of Genetics with Sciences of the Past in Croatia” is a 3-year project linking the expertise of the University of Cambridge and the University of Pisa with the University of Zagreb to research the rich, yet-to-be-fully-explored heritage of the eastern Adriatic region.
Dr Preston Miracle, lead archaeologist of the Cambridge contingent of the project, said, “The potential cultural heritage of the region is enormous, ranging through the full spectrum of human occupation from the Palaeolithic to present day. The scientific potential of such material can only be reached through the use of techniques and methodologies in which the partner organisations have great expertise.”
The “Mend the Gap” project was ranked No 1 of 65 funded projects (out of a total of 546 submissions) from all research fields and parts of the EU. (No other projects with a Cambridge link were successful.) As defined by the European Commission, Twinning projects “will help strengthen a defined field of research in a knowledge institution through linking with at least two internationally-leading counterparts in Europe.”
By providing access to the scientific and administrative expertise and experience, both Italy (Pisa) and the UK (Cambridge), as partners in this Twinning exercise, will enable scientists at the host institution (Zagreb) to increase their capacity to obtain national, European and international (global) funding for research in a region which has much to offer to both the archaeological and scientific world in general.
The Eastern Adriatic holds a large number of important archaeological sites. Remains have been found and identified, although they are far from being analysed, exploited and/or commercialised to their full capacity. These Croatian archaeological sites are of outstanding universal value from the historical, aesthetic, ethnological, anthropological and/or educational points of view. Moreover, the information which can be retrieved when analysing these sites can help us to solve many of the problems currently faced. At the same time, they can stimulate the regional scientific community and add to the development of local communities.
A good example of an important archaeological site is Vela Spila, situated above the town of Vela Luka on the Korčula Island. Although only a small portion of the site has been excavated and analysed to this day, results confirm that this is one of the richest and most promising Eastern Adriatic archaeological sites. There are a few, out of many, significant finds yielding from Vela Spila important to single out due to their extraordinary nature. One of the most significant finds are 36 ceramic figurines, representing the first evidence of ceramic figurative art in late Upper Palaeolithic Europe, c. 17,500–15,000 years before present.
To illustrate the importance of these figurines in a broader context, it is important to mention that there are only two other ceramic figurine-bearing European Upper Palaeolithic sites, both of which are situated in Central Europe, with Vela Spila being the single Mediterranean example. In addition, Vela Spila contains directly dated Mesolithic burials of known archaeological context, both juvenile and adult specimens. Human remains, directly dated to the Mesolithic period, are scantily found across Europe, and Vela Spila Mesolithic specimens represent a rare discovery in Eastern Mediterranean context.
ArchaeoLink will be working with the Mend the Gap team by liaising between the archaeologists and the community of Vela Luka in which is situated the site of Vela Spila. Their purpose is to both help promote the archaeological research and to assist the community to obtain educational, social and economic benefits from the site. This ensures that the impact is not only relevant but is also beneficial. | https://www.croatiaweek.com/cambridge-archaeologists-secure-e1m-eu-grant-for-3-year-croatia-project/ |
A novel and simply calculated nutritional index serves as a useful prognostic indicator in patients with coronary artery disease.
No nutritional index has been firmly established yet in patients with coronary artery disease (CAD). In this study, we propose a simple to calculate nutritional indicator in patients who underwent percutaneous coronary intervention (PCI) by using parameters routinely measured in CAD and evaluated its prognostic implication. This study is a retrospective observational analysis of a prospective database. The subjects were consecutive 3567 patients underwent their first PCI between 2000 and 2013 at Juntendo University Hospital in Tokyo. The median of the follow-up period was 6.3 years (range: 0-13.6 years). The novel nutritional index was calculated by the formula; Triglycerides (TG) × Total Cholesterol (TC) × Body Weight (BW) Index (TCBI) = TG × TC × BW / 1000 (TG and TC: mg/dl, and BW: kg). The Spearman non-parametric correlation coefficient between TCBI and the most often used conventional nutritional index, Geriatric Nutritional Risk Index (GNRI), was 0.355, indicating modest correlation. Moreover, Unadjusted Kaplan-Meier analysis showed higher all-cause mortality, cardiovascular mortality, and cancer mortality in patients with low TCBI. Consistently, elevation of TCBI was associated with reduced all-cause (hazard ratio: 0.86, 95%CI: 0.77-0.96, p < 0.001), cardiovascular (0.78, 0.66-0.92, p = 0.003), and cancer mortality (0.76, 0.58-0.99, p = 0.041) in patients after PCI by multivariate Cox proportional hazard analyses. TCBI, a novel and easy to calculate nutrition index, is a useful prognostic indicator in patients with CAD.
| |
7 Gretchen Helmke and Steven Levitsky, 'Informal Institutions and Comparative Politics: A Research Agenda', Perspectives on Politics, 2 (December 2004): 725–40.
8 See Lauth, Gryzmala-Busse and Luong, and Helme and Levitsky on this, as well as the discussion in Chapter 2. policy processes in each country. In Bosnia, the High Representative directly participated in the policy-making process by establishing the reform commissions, appointing their chairs and members and setting the rules of procedure. Paddy Ashdown chose which of the PRC conclusions to follow, while Miroslav Lajčak came up with his own proposal for police reform after previous versions had not been accepted by the RS parliament. The High Representative was appointed as the final arbiter of the Dayton provisions and their meanings. Their role in Bosnian politics is to ensure that Dayton agreement is respected by all sides and any resistance to it is removed. With such powers at their disposal, the HRs is an important actor in the policy process in Bosnia, and their behaviour affects the actions of the domestic actors.
Even though he did not refrain from putting pressure on RS politicians to initiate reforms, during military reforms in Bosnia Ashdown largely fulfilled his role in a neutral fashion, which enabled domestic politicians to find a compromise. To the extent that he had a preference, he represented the interest of the wider international community, on behalf of whom he was appointed as HR. Similarly to NATO representatives, for example, he spoke about the need to centralise and cut the size of the army. However, during police reform, he was perceived by RS politicians to be acting in a biased manner, first by refusing to instruct the PRC to use consensus and then by insisting on a reform proposal that the Serb leaders found unacceptable. Along with a lack of coordination between him and the EU in meeting the requirements for successful police reform, this inhibited compromise instead of enabling it. Later on, when Miroslav Lajčak tried to amend the quorum rules in the BIH Parliament, the HR's reputation suffered further. His actions to relax decision-making rules were seen as an attempt to scrap veto mechanisms that had given minority groups protection from outvoting. The HR's failure in this case to act as a credible and unbiased factor in the policy process further contributed to the failure of police reform.
In Macedonia, there is no similar figure to the High Representative, so the direct involvement of external actors in the policy process is more limited. Most of the external input in domestic politics is provided through the indirect, advisory and facilitating efforts of international organisations and the ambassadors of powerful countries and organisations, such as the US and the EU. In both countries, statements, criticisms and visits from high-ranking international officials tend to receive a lot of domestic attention and can affect the behaviour of domestic politicians. This indirect influence of external actors has generally proved helpful in encouraging ethnic accommodation.
NATO and EU officials used membership conditionality to elicit greater willingness for accommodation among the political leadership. In Macedonia, when the initial law on local government was debated in 2002, its adoption was set as a condition by EU and NATO representatives for holding the donors' conference that would raise funds for post-conflict reconstruction. In 2004, the EU firmly took the government's side in supporting the new territorial organisation and encouraging people not to vote on the referendum over municipal boundaries. In both instances, conditionality resulted in the expected outcomes, although in the case of the referendum it was aimed at the population rather than the politicians. In Bosnia, conditionality was applied in both police and military reform, but was only successful in the first case. The difference between the two cases was that with military reform NATO had a clear and consistent set of conditions, which neither changed over time nor were they tailored specifically for the country's accession to the Partnership for Peace programme. The EU had no prior criteria for police restructuring and introduced its three conditions specifically for Bosnia. It eventually went back on all of them, demonstrating that the conditions applied were neither consistent nor fair. Credible and consistent conditionality therefore produced better results in encouraging accommodation, as the rewards and punishments administered to political elites were more predictable and reliable.
The case of police reform in Bosnia also illustrates the limits of conditionality as an instrument for propelling domestic reforms. Contrary to the claims of the literature on the 'transformative power' of EU and NATO, conditionality can sometimes fail to trigger the desired transformation in the receiving state.9 The case of Bosnian police reform shows that this process can run in both directions and instead of EU or NATO integration efforts working to de-ethnicise domestic issues, ethnic divisions can spill into EU and NATO integration, making them ethnically divisive issues. RS politicians certainly resented the role that the EU played during the repeated police reform attempts. Their trust in EU integration and their commitment to the EU integration process has waned as a result. That can undermine the commitment to EU and NATO membership as a shared goal among all groups in society, and can remove a powerful lever that those organisations have at their disposal. Moreover, ethnicisation of EU or NATO integration removes a significant potential cross-cutting cleavage in politics that could otherwise re-frame policy debates in terms other than gains and costs for ethnic groups. | https://m.ebrary.net/1466/political_science/external_actors |
Gnosis is a Greek word which means ‘insight’, an intuitive process of knowing oneself. And to know oneself, is to know human nature and human destiny.
All religious traditions acknowledge the world is imperfect. Few are able to account for all the pain and suffering that exists in our world.
Gnostics believe the world is flawed because it was created in a flawed manner. Gnostics believe there is a true, ultimate and transcendent God, who is beyond all created universes and never created anything. The basic gnostic myth refers to Aeons, intermediate deific beings who exist between the ultimate true God and ourselves. The Aeons and the true God comprise the realm of Pleroma (Fullness), the totality of divine powers.
Sophia (Wisdom) is one of the aeons who, from her own volition, emanated a flawed consciousness called the Demiurge. The Demiurge then created Earth and all living things which are also flawed. In effect, humanity is the product of a cosmic error!
The Demiurge was also the creator of the Archons or rulers to keep humanity in order.
Human nature mirrors the duality found in the world; consisting of a perishable physical component made by the false God, the Demiurge, and a fragment of the divine essence from the true God, referred to as the divine spark, or the I AM. The gnostic’s mission is to liberate that divine spark.
Human beings are often unaware of this divine spark within them. The Demiurge and his Archons are intent upon keeping men and women ignorant of their true nature and destiny. All our earthly attachments keep us enslaved to these lower cosmic rulers. Death releases the divine spark, but unless the individual has attained gnosis prior to death, they will reincarnate back into the physical world.
Fortunately, Messengers of Light, or Avatars, have been sent from the true God to assist us in our quest for gnosis. Jesus is considered the main saviour figure by gnostics.
In the gnostic texts Jesus speaks of illusion and enlightenment, not of sin and repentance. Instead of redeeming us from our sins, he comes as a guide to aid our spiritual understanding. When the individual attains enlightenment, Jesus is no longer their spiritual master – they become equal.
Gnosticism encourages us to become a ‘being in the world, but not of it’, a lack of egotism, and a respect for the freedom and dignity of all other beings.
In the fullness of time, with conscious effort, every spiritual being will be reunited with its higher self and be able to enter the Pleroma.
Most gnostic texts are written in mythical form to convey a deeper, underlying meaning. A myth is not only a story, it is a statement made in symbols. The language of the unconscious is symbolic. A symbol speaks directly to the soul, even when the mind does not understand. When a symbol touches the soul there is a change. Every myth takes its origin in inspiration, from the Latin inspirare, a message from the spirit, the breath of life. The spirit speaks from the cosmos, and gives us answers to the unanswered. Meeting this spirit fulfils a destiny way beyond our material world and these texts are trying to show us the way.
The gnostics believe Jesus only imparted advance teachings to the spiritual elite while withholding them from the uninitiated.
In the Gospel of Thomas Jesus said, “I disclose my mysteries to those who are worthy of my mysteries.”
In my next blog, I will be discussing The Secret Book of John, considered to be the most important and valued gnostic text from the Nag Hammadi Library. | https://marymagdaleneslegacy.com/2021/03/08/the-gnostic-view-of-creation/ |
This schedule should be read in conjunction with the engagement letter and the standard terms and conditions.
PERSONAL TAX – SOLE TRADERS AND PROPERTY INCOME
Recurring compliance work – accounts
1 We will prepare the business accounts in accordance with generally accepted accounting standards from the books, accounting records and other information and explanations provided to us by you and/or by others on your behalf.
2 Where receipts for your property business are less than £150,000 for the tax year we will prepare property accounts on the default cash basis unless we agree with you that it is appropriate to elect to use the accruals basis and the accounts are to be prepared on that basis. We will then deal with the election on completion of your tax return.
3 We will complete the writing up of your books and records in so far as they are incomplete when presented to us. These will be from the accounting information and records you supply.
4 We will not be carrying out any audit work as part of this assignment and accordingly will not verify the assets and liabilities of the business, nor the items of expenditure and income. To carry out an audit would entail additional work to comply with International Standards on Auditing so that we could report on the truth and fairness of the financial statements. Accordingly, we shall not seek any independent evidence to support the entries in the accounting records, or to prove the existence, ownership or valuation of assets or completeness of income, liabilities or disclosure in the accounts. Nor shall we assess the reasonableness of any estimates or judgements made in the preparation of the accounts. Consequently, our work will not provide any assurance that the accounting records are free from material misstatement, irregularities or error. We would also like to emphasise that we cannot undertake to discover any shortcomings in your systems or irregularities on the part of your employees.
5 We have a professional duty to compile accounts that conform with generally accepted accounting principles. Where we identify that the accounts do not conform with generally accepted accounting principles or standards, we will inform you and suggest amendments be put through the accounts before being finalised. We have a professional responsibility not to allow our name to be associated with accounts that may be misleading. In extreme cases, where this matter cannot be resolved, we will withdraw from the engagement and notify you in writing of the reasons.
6 Should you instruct us to carry out any alternative report it will be necessary for us to issue a separate letter of engagement.
7 To ensure that anyone reading the accounts is aware that we have not carried out an audit, we will attach to the accounts a report stating this fact.
8 There are no third parties that we have agreed should be entitled to rely on the work done pursuant to this engagement letter.
Recurring compliance work – tax
9 We will prepare your self-assessment tax returns together with any supplementary pages required from the information and explanations that you provide to us. After obtaining your approval, we will submit your returns to HMRC.
10 We will calculate your income tax, high-income child benefit charge, national insurance contributions (NIC) and any capital gains tax liabilities as included on your self-assessment return, and tell you how much you should pay and when. Where instructed by you, we will advise on the interest and penalty implications if tax or NIC is paid late. We will also check HMRC’s calculation of your tax and NIC liabilities, and initiate repayment claims if tax or NIC has been overpaid.
11 Other than tax credits and universal credit (see below), we will advise you as to possible tax return-related claims and elections arising from information supplied by you. Where instructed by you, we will make such claims and elections in the form and manner required by HMRC.
12 We will review PAYE notices of coding provided to us by you and advise accordingly. Note that HMRC no longer sends copies of notices of coding to agents.
Ad hoc and advisory work
13 Where you have instructed us to do so we will provide such other taxation ad hoc and advisory services as may be agreed between us from time to time. These services will be subject to the terms of this engagement letter and standard terms and conditions of business unless we decide to issue a separate engagement letter. An additional fee may be charged for these services. Examples of such work include:
- advising on the in-year Capital Gains Tax (CGT) reporting requirements on disposals of property, and preparing the in-year return and calculating the CGT due where required. We will require you to provide information as early as possible in advance of exchange of contracts in order to provide advice on the tax implications, reporting requirements and to quantify the tax bill;
- advising on ad hoc transactions (for example, pre-sale advice on the sale of assets) and queries (including telephone conversations), preparing and submitting information in the relevant format to HMRC and calculating any related tax liabilities;
- advising on double tax relief if appropriate;
- dealing with any enquiry opened into your tax return or tax affairs by HMRC;
- preparing any amended returns that may be required and corresponding with HMRC as necessary; and
- advising on the rules relating to and assisting with VAT registration.
14 Where specialist advice is required, on occasions we may need to seek this from or refer you to appropriate specialists. We will only do this when instructed by you.
Tax credits and universal credit
15 If we agree to advise you on tax credits and universal credit we will issue a separate letter or schedule to cover this area. Tax credits and universal credit are, in effect, a social security benefit. Your entitlement or otherwise will depend not only on your own circumstances but also those of your household, and we would require all relevant information to advise in this regard.
Changes in the law or practice or in public policy
16 We will not accept responsibility if you act on advice given by us on an earlier occasion without first confirming with us that the advice is still valid in the light of any change in the law or practice or in public policy or your circumstances.
17 We will accept no liability for losses arising from changes in the law or practice or in public policy that are first published after the date on which the advice is given.
Your responsibilities
18 It is your responsibility to keep proper accounting records that disclose with reasonable accuracy at any particular time the financial position of the business. It is also your responsibility to safeguard the assets and to take reasonable steps for the prevention of and detection of fraud and other irregularities with an appropriate system of internal controls.
19 You are also responsible for making available to us, as and when required, all accounting records and all other relevant records and related information, including minutes of meetings.
20 You will also be responsible for:
a maintaining records of all receipts and payments of cash;
b maintaining records of invoices issued and received;
c reconciling balances annually with the bank statements;
d preparing a record of business mileage undertaken in the year;
e preparing a record of hours per month worked at home if you wish to claim for business use of your home;
f preparing details of any loan interest paid;
g preparing a record of capital introduced; and
h preparing details of the following at the year-end: stocks and work in progress, fixed assets, amounts owing to creditors, amounts owing by customers, and accruals and prepayments.
21 You are legally responsible for:
a ensuring that your self-assessment tax returns are correct and complete;
b filing any returns by the due date; and
c paying tax on time.
Failure to do this may lead to penalties and/or interest.
22 Taxpayers who approve their returns cannot delegate this legal responsibility to others. You agree to check that returns that we have prepared for you are complete before you approve them.
23 To enable us to carry out our work you agree:
a that all returns are to be made on the basis of full disclosure of all sources of income, charges, allowances and capital transactions;
b to provide all information necessary for dealing with your affairs, including the accounts information outlined above: we will rely on the information and documents being true, correct and complete, and will not audit the information or those documents;
c to provide full details of all UK residential property disposals including associated costs/valuations prior to disposal. Where you consider that you will be non-UK resident in the tax year of disposal, full details of all UK property disposals (residential and non-residential) and disposals of shares in UK property rich companies or UK property rich collective investment vehicles, or such other entities must be advised prior to exchange of contracts or agreement to transfer shares or units. If information is received after this we cannot guarantee that we can provide advice on the amount of capital gains tax due or submit an in-year return within 30 days after the completion date of the disposal.
d to authorise us to approach such third parties as may be appropriate for information that we consider necessary to deal with your affairs; and
e to provide us with information in sufficient time for your tax return to be completed and submitted by the due date following the end of the tax year. In order that we can do this, we need to receive all relevant information by 30th September. Where feasible, we may agree to complete your return within a shorter period but may charge an additional fee for so doing.
24 You will keep us informed of material changes in your circumstances that could affect your tax liability. If you are unsure whether the change is material or not please let us know so that we can assess its significance.
25 Where you wish us to deal with HMRC communications you will forward to us all communications received from HMRC such as HMRC statements of account, copies of notices of assessment, tax codes and letters. These must be provided in time to enable us to deal with them as may be necessary within the statutory time limits. It is essential that you let us have copies of any correspondence received because HMRC is not obliged to send us copies of all communications issued to you.
26 (a) If you provide digital services to consumers in the European Union you are responsible for either registering for VAT in that member state or registering for VAT Mini One Stop Shop (MOSS).
b You are responsible for monitoring your monthly turnover to establish whether you are liable to register for VAT. If you do not understand what you need to do, please ask us. If you exceed the VAT registration threshold and wish us to assist you in notifying HMRC of your liability to be VAT registered, we will be pleased to assist you in the VAT registration process. You should notify us of your instructions to assist in your VAT registration in good time to enable a VAT registration form to be submitted within the time limit of one month following the month in which you exceed the VAT registration threshold in force at that time. We will not be responsible if you fail to notify us in time and incur a late registration penalty as a result.
c You are responsible for employment taxes, pensions (including auto-enrolment) and the assessment of the tax status of your workers. If you do not understand what you need to consider or action you need to take, please ask us. We will not be in a position to assist you in complying with your responsibilities if we are not engaged to provide such a service. We are not responsible for any penalty that is incurred.
27 Our services as detailed above are subject to the limitations on our liability set out in the engagement letter and in paragraph 18 of our standard terms and conditions of business. These are important provisions that you should read and consider carefully. | https://skryba.co.uk/en/self-employed |
Woodland Hills car accident attorney Barry P. Goldberg has written several articles regarding the use of Code of Civil Procedure section 998 Offers to Compromise. In fact, his west San Fernando Valley personal injury law office uses 998 Offers to Compromise in every case in litigation, including uninsured and underinsured motorist arbitrations in order to shift the costs of litigation to the insurance companies. In some manner, every article from the Goldberg office has advised counsel to serve the 998 Offer to Compromise in the exact proper format— 1. Capable of Acceptance; and 2. Specific as to terms and singular as to party.
A great amount of confusion has arisen in California and with actively practicing lawyers regarding the use of CCP 998 Offers to Compromise either to customize a settlement or to prompt settlement discussions by making the Offer “all inclusive.” Although it may be true that an insurer may be required to respond to settlement overtures in both the third party and first party contexts, to actually shift costs, the CCP 998 statutory rules must be religiously followed. In fact, in most Offers to Compromise, defendants want to avoid the entry of a judgment.
In the recent case of Mostavi Law Group, APC v. Larry Rabineau (2nd Dist, March 3, 2021) the Court was faced with the precise problem that has arisen over the last several years of counsel failing to serve CCP 998 Offers to Compromise in the exact proper form. In this case, the defendant served an Offer to Compromise. However, The Offer did not specify how plaintiff could accept it. Accordingly, plaintiff’s counsel hand wrote an acceptance onto the offer itself and filed the notice of acceptance with the court. The court entered judgment pursuant to 998 (b)(1).
Defendant moved to vacate the judgment arguing that his own Offer to Compromise was invalid because it lacked an acceptance provision. The trial court agreed and set aside the judgment. Although there is some conflicting case law and enforcing a settlement is consistent with the section’s policy and purpose, the court of appeal affirmed. The Court went through a complete and in-depth analysis of section 998.
The Court pointed out that the Legislature enacted Code of Civil Procedure section 998 to encourage and expedite settlement of lawsuits before trial. To effectuate this purpose, the statute simultaneously promotes the extension and acceptance of reasonable pretrial offers to compromise. The “policy is plain. It is to encourage settlement by providing a strong financial disincentive to a party—whether it be a plaintiff or a defendant—who fails to achieve a better result than that party could have achieved by accepting his or her opponent’s settlement offer. (This is the stick. The carrot is that by awarding costs to the putative settler the statute provides a financial incentive to make reasonable settlement offers.)” (Bank of San Pedro v. Superior Court (1992) 3 Cal.4th 797, 804.)
However, the code has specific instructions. Section 998 (b) requires, among other things, that a party seeking to take advantage of the statute serve on an opposing party a written offer to have judgment entered on specified terms. Most important, for purposes of this appeal, the written offer “shall” contain what has come to be known as an “acceptance provision. Specifically, the statute states that the written offer “shall” include “a provision that allows the accepting party to indicate acceptance of the offer by signing a statement that the offer is accepted.” (§ 998 (b), emphasis added.) It provides:
“Not less than 10 days prior to commencement of trial . . . , any party may serve an offer in writing upon any other party to the action to allow judgment to be taken or an award to be entered in accordance with the terms and conditions stated at that time. The written offer shall include a statement of the offer, containing the terms and conditions of the judgment or award, and a provision that allows the accepting party to indicate acceptance of the offer by signing a statement that the offer is accepted. Any acceptance of the offer, whether made on the document containing the offer or on a separate document of acceptance, shall be in writing and shall be signed by counsel for the accepting party[.]”
“If the offer is accepted, the offer with proof of acceptance shall be filed and the clerk or the judge shall enter judgment accordingly.” (§ 998, subd. (b)(1).) However, “[i]f an offer made by a defendant is not accepted and the plaintiff fails to obtain a more favorable judgment or award, the plaintiff shall not recover his or her postoffer costs and shall pay the defendant’s costs from the time of the offer.” (§ 998, subd. (c)(1).) The trial court also has discretion to “require the plaintiff to pay a reasonable sum to cover postoffer costs of the services of expert witnesses[.]”
Applying the statute to the facts, the Court did not find that there was a binding contract and that equitable principles would not establish an equitable estoppel. This is important for those law firms that use the section 998 Offers to Compromise for its true purpose— to shift litigation costs— as opposed to merely a settlement overture. Based on the foregoing, we recommend that the 998 Offer to Compromise not deviate from the exact form and that the Offer be capable of acceptance. Do not copy the defendant/insurer’s form. The insurers do not want a judgment entered against its own insured. | https://barrypgoldberg.com/ccp-998-offer-compromise-must-proper-form/ |
Q:
what is an inverse femto barn?
I came across the use of the unit barn and inverse barn while reading about the operation of LHC. What is an inverse femtobarn ? What does it tell about the experiment being described ?
A:
A barn is a unit of area, equal to $10^{-28}\text{ m}^2$. The prefix femto- signifies $10^{-15}$, so a femtobarn is equal to $10^{-43}\text{ m}^2$. When high-energy physicists talk about inverse femtobarns, they mean collisions per femtobarn of beam cross-sectional area. The inverse femtobarn is not a measure of information, but rather of the effectiveness of the particle accelerator.
Here's the reasoning behind that: imagine two beams of particles coming at each other, and for simplicity let's designate one beam as the "target" and the other as the "probe" (it doesn't matter which is which). The scattering cross section $\sigma$ is the effective size of a single target particle, as "seen" by a probe. So the probability of a collision will be just the fraction of the target beam area that is actually occupied by the target particles; in other words, the scattering cross section divided by the average cross-sectional area per particle,
$$P_\text{collision} \approx \frac{\sigma}{A_t/n_t}$$
($t$ for "target"). Here I'm assuming that the beam is highly diffuse, and that we're only talking about one "bunch" (a finite-length section of a particle beam).
The total number of collisions is just this probability times the number of chances for a collision to happen, namely the number of probe particles in a bunch:
$$n_\text{collision} = \sigma \frac{n_t n_p}{A_t}$$
($p$ for "probe"). Finally, since this happens every time two bunches cross paths, to get the overall rate at which collisions occur you need to multiply by the rate of bunch crossings, which is denoted $f$.
$$R_\text{collision} = \sigma \frac{f n_t n_p}{A_t}$$
The quantity $\frac{f n_t n_p}{A_t}$ is defined as the two-beam luminosity $L$. Roughly speaking, it represents the number of potential collisions per unit area per unit time.
To find the actual number of collisions for any particular process, you multiply by luminosity by the cross-section for that process and integrate over time,
$$n_{X\to Y} = \bigl(\sigma_{X\to Y}\bigr)\times\biggl(\int L \,\mathrm{d}t\biggr)$$
This split is convenient because the first factor depends only on the physical process being considered, and the second factor depends only on the design of the accelerator. So that second factor is a good way to characterize the production capacity of a given accelerator. The units of this value are 1/area, and the inverse femtobarn turns out to be roughly the right magnitude for measuring $\int L \,\mathrm{d}t$ at current particle accelerators. This is what particle physicists mean when they talk about inverse femtobarns.
| |
bradley started his professional career in studio photography in 2008 taking product shots for his clients he has since expanded his passion to include weddings, real estate, portraits, landscapes, photo retouching and restoration along with teaching others photography and editing skills. Using this knowledge to create images that capture the essence of each property he is shooting. This love for what he does has lead him to expand his photoshop editing skills to include virtually staging a vacant room, also adding measuring and drawing of floor plans for many of his clients. | http://bephotography.biz/about.html |
Experts in Cuba have undertaken the initialization of a new plant based on Japanese technology to deal with the destruction of substances that cause Ozone Layer depletion.
The fragile gaseous strata filters sunlight and impedes harmful solar ultraviolet radiation reaching the surface of the Earth, thus preserving human, plant and animal life.
Natacha Figueredo MSc, specialist from the Ozone Technical Office (OTOZ in Spanish) explained to the Havana Reporter that this modern installation constructed in the Siguaney cement factory in the province of Sancti Spiritus, commenced operations last April,and is presently in a functional stabilization phase.
During the first stage, Ozone depletion substances (SAO in Spanish) collected during the substitution of more Cuba Eliminates Substances that Deplete Ozone Layer than 2,500,000 refrigerators and almost 300,000 air conditioners in the residential sector are to be destroyed.
The works form part of the “Energy Revolution” which fully eliminated the use of chlorofluorocarbons (CFC) in Cuba in domestic refrigeration.Hydrofluorocarbons (HCFC) will later be destroyed in the plant which will, over the coming months, be collected from refrigeration and air conditioning units around the country.
Through this initiative, Cuba has attained the destruction this year of some 258.4 kilos of SAO, a result which places the island within an elite group ofnations in the region with the capability to undertake this complex process.
The installation is part of a demonstrative collection, recovery, storage, transport and regeneration ofsubstances detrimental to the ozone layer initiative, that is the result of a strategy developed by the OTC and the Montreal Multilateral Protocol Fund, via the United Nations Development Program(PNUD in Spanish).
The project seeks to ensure an environmentally safe outcome to SAO destruction by averting emission into the earth’s atmosphere, thus contributing to Cuba meeting Montreal Protocol Commitments to gradually eradicate and reduce SAO use.
Cuba is the first country to totally eliminate CFC consumption in domestic refrigeration, a significant contribution to the confrontation of climate changerelated issues that affect the planet, because the gases that have an impact on the Ozone Layer have a potent greenhouse effect. According to OTOZ data, the actions undertaken on the island have reduced CO2 atmospheric emissions by 4 million tons per year.
OTOZ director and doctor of Sciences, Nelson Espinosa explained that one of the most notable Cuban achievements of the past twenty years is the total elimination of a group of substances that deplete the Ozone layer, including the use of CFC’s in the manufacture of pharmaceutical and industrial aerosols, and methyl bromide in the fumigation of crops, storage units and other industrial installations. | http://havanareporternews.com/health_and_medicine/cuba-eliminates-substances-deplete-ozone-layer.html |
A prospective cohort study of 122 adult patients presenting to an otolaryngologist's office with globus pharyngeus.
To investigate the epidemiology of globus pharyngeus in adult patients presenting to the otolaryngologist's office. Also, the predictors of persisting symptoms, prevalence of anxiety and the effect of clinical assessment were analysed. This was a prospective cohort study. Follow-up was carried out using a postal questionnaire. One otolaryngologists' office comprising three medical doctors. A total of 122 consecutive globus patients presenting to one otolaryngology office in a 1-year period. Globus incidence, gender and age distribution, predictors of persisting symptoms and the patient's health-related concerns. 3.8% of first-time visits were regarding globus. The mean age was 48 years [range 20-88 years], and a female predominance was found (ratio 1.49). Eighty-four per cent experienced anxiety, mainly due to fear of cancer. The most common pathological findings were reflux (15.6%) and post-infectious inflammation (10.6%). 21.4% of questionnaire responders reported full remission of their symptoms. Three predictors regarding symptom persistence were identified: male gender (OR 1.52), smoking (OR 3.4) and difficulties in breathing (OR 8.7). Patients with concomitant foreign body sensation were less likely to have persisting symptoms (OR 0.42). No cases of malignant disease were encountered. 94.7% was reassured by the office visit. The incidence of globus is 3.8% in the otolaryngologist's office. Female gender and concomitant foreign body sensation were predictive for presenting to the clinic even if symptom remission had occurred. Male gender, smoking and self-perceived breathing difficulties were predictive for persisting symptoms. Globus is an anxiety-causing symptom, but reassurance is provided by clinical examination by the otolaryngologist.
| |
CROSS-REFERENCE TO RELATED PATENT APPLICATIONS
Japan Priority Application 10-129486, filed Apr. 23, 1998 including the specification, drawings, claims and abstract, is incorporated herein by reference in its entirety.
BACKGROUND OF THE INVENTION
SUMMARY OF THE INVENTION
The present invention relates to a product sum operation device and, more particularly, to a product sum operation device that is capable of carrying out a fast product sum operation with real and complex numbers.
With recent increase in capacity and function in the communication industry, faster processing becomes more and more important for modulation/demodulation, Fourier transform, and base band transmission. These types of processing uses a product sum operation with the real and complex numbers. For the product sum operation with complex numbers, a single operation involves in four-time multiplication, three-time addition, and one-time subtraction of real numbers. A larger sampling frequency results in a shorter processing time available for a single product sum operation with the real numbers. Thus such a product sum operation device is essential that is capable of carrying out an operation with real and complex numbers at a higher speed.
The product sum operation device for a faster product sum operation is disclosed in, for example, Japanese Patent Laid-open No. 9-269939. As will be described more in detail below, the product sum operation device disclosed expedites the product sum operation with the real and complex numbers by means of two parallel processing of multiplication and addition. However, the increase in speed to this extent may not be sufficient to provide a satisfactory performance with a possible larger capacity required for communications in the future. Furthermore, the product sum operation device disclosed has a rather complex configuration of an input section including two delay elements and two multiplexers.
Therefore, an object of the present invention is to provide a product sum operation device that is capable of carrying out a fast product sum operation with real and complex numbers.
Another object of the present invention is to provide a product sum operation device having a relatively simple configuration.
A product sum operation device according to the present invention achieves a fast product sum operation with real and complex numbers by means of increasing the number of arithmetic operation units and data transfer units to thereby increase the level of parallelism.
The product sum operation device according to a first aspect of the present invention comprises a data transfer section for transferring four data simultaneously; and first through fourth multipliers adapted to receive two data that are assigned in advance out of the four data transferred from the data transfer section to carry out a multiplication operation with the two data received. The product sum operation device further comprises a first adder-subtracter for carrying out adding and subtracting operations with the multiplication results obtained by the first and second multipliers; a second adder-subtracter for carrying out adding and subtracting operations with the multiplication results obtained by the third and the fourth multipliers; and first and second accumulators. The product sum operation device further comprises a first adder for adding the addition result obtained by the first adder-subtracter to a value previously stored in the first accumulator to give the addition result to the first accumulator as a new value; and a second adder for adding the addition result obtained by the second adder-subtracter to a value previously stored in the second accumulator to give the addition result to the second accumulator as a new value.
A product sum operation device according to a second aspect of the present invention comprises a data transfer section for transferring four data simultaneously; and first through fourth multipliers provided correspondingly to one of the four data transferred from the data transfer section to calculate square of the data transferred. The product sum operation device further comprises a first adder-subtracter for carrying out adding and subtracting operations with the multiplication results obtained by the first and second multipliers; a second adder-subtracter for carrying out adding and subtracting operations with the multiplication results obtained by the third and the fourth multipliers; and first and second accumulators. The product sum operation device further comprises a first adder for adding the addition result obtained by the first adder-subtracter to a value previously stored in the first accumulator to give the addition result to the first accumulator as a new value; and a second adder for adding the addition result obtained by the second adder-subtracter to a value previously stored in the second accumulator to give the addition result to the second accumulator as a new value.
A product sum operation device according to a third aspect of the present invention comprises first through fourth buses for transferring first through fourth data; and first through fourth multiplexers connected to two buses out of the first through the fourth buses that are assigned in advance. The first through the fourth multiplexers are for selecting, in response to a selection signal, one of the data transferred over the two buses in accordance with the selection signal to produce selected data. The product sum operation device further comprises a first multiplier for multiplying the data transferred over the first bus with the data supplied from the first multiplexer; a second multiplier for multiplying the data transferred over a second bus with the data supplied from a second multiplexer; a third multiplier for multiplying the data transferred over a third bus with the data supplied from a third multiplexer; and a fourth multiplier for multiplying the data transferred over the fourth bus with the data supplied from the fourth multiplexer. The product sum operation device further comprises a first adder-subtracter for carrying out adding and subtracting operations with the multiplication results obtained by the first and the second multipliers; a second adder-subtracter for carrying out adding and subtracting operations with the multiplication results obtained by the third and the fourth multipliers; and first and second accumulators. The product sum operation device further comprises a first adder for adding the addition result obtained by the first adder-subtracter to a value previously stored in the first accumulator to give the addition result to the first accumulator as a new value; and a second adder for adding the addition result obtained by the second adder-subtracter to a value previously stored in the second accumulator to give the addition result to the second accumulator as a new value.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1
is a block diagram showing a configuration of a conventional product sum operation device;
FIG. 2
is a block diagram showing a configuration of a product sum operation device according to a first embodiment of the present invention;
FIG. 3
is a block diagram showing a configuration of a product sum operation device according to a second embodiment of the present invention; and
FIG. 4
is a block diagram showing a configuration of a product sum operation device according to a third embodiment of the present invention.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
FIG. 1
FIG. 1
41
42
43
46
49
50
43
46
49
50
44
47
49
50
45
48
51
52
53
54
55
56
51
52
49
50
53
54
Referring to , a conventional product sum operation device is described for the purpose of facilitating the understanding of the present invention. In , the product sum operation device comprises buses and for receiving data, delay elements and , and multipliers and . The delay elements and are for holding data received in the latest step. The multipliers and carry out a multiplication operation with the received data. Multiplexers and are for selecting inputs to the multipliers and in accordance with control signals and . The product sum operation device further comprises adder-subtracters and , accumulators and , and buses and for producing data. The adder-subtracters and are for adding or subtracting cumulatively the multiplication results obtained by the multipliers and to or from the values held by the accumulators and .
This product sum operation device may be used for the implementation of an FIR filter and for the product sum operation with complex numbers. These applications are now described. First, implementation of the FIR filter is described for the case of a fourth order filter.
A filter output y(n) at a time instant n can be given by the equation (1), y(n) equals the product of a0 and x(n) plus the product of a1 and x(n−1) plus the product of a2 and x(n−2) plus the product of a3 and x(n−3), that is:
&lt;PTEXT&gt;&lt;PDAT&gt;y(n)&equals;a0&middot;x(n)&plus;a1&middot;x(n&minus;1)&plus;a2&middot;x(n&minus;2)&plus;a3&middot;x(n&minus;3)&emsp;&emsp;(1)&lt;/PDAT&gt;&lt;/PTEXT&gt;
The operation device carries out calculation operations in parallel for the filter output y(n) at the time instant n that is given by the equation (1) and for a filter output y(n+1) at a time instant (n+1) given by the equation (2), y(n+1) equals the product of a0 and x(n+1) plus the product of a1 and x(n) plus the product of a2 and x(n−1) plus the product of a3 and x(n−2), that is:
&lt;PTEXT&gt;&lt;PDAT&gt;y(n&plus;1)&equals;a0&middot;x(n&plus;1)&plus;a1&middot;x(n)&plus;a2&middot;x(n&minus;1)&plus;a3&middot;x(n&minus;2)&emsp;&emsp;(2)&lt;/PDAT&gt;&lt;/PTEXT&gt;
In the above equations (1) and (2), x(n+1), x(n), x(n−1), x(n−2), and x(n−3) are filter inputs received by the filter at time instants (n+1), n, (n−1), (n−2), and (n−3), respectively. In addition, a0, a1, a2, and a3 are filter coefficients.
The operation device requires five steps for the calculation with the equations (1) and (2). It is noted that only a last fifth step is described here but first through fourth steps are similar in operation to the fifth step except for the data subjected to the calculation.
53
54
46
Before initiation of the fifth step, the accumulator holds the calculation result of a1·x(n)+a2·x(n−1)+a3·x(n−2) in the operation of the fourth step. On the other hand, the accumulator holds the calculation result of a1·x(n−1)+a2·x(n−2)+a3·x(n−3). The delay element is supplied with the filter input x(n).
41
42
46
41
The fifth step is then initiated. First, the filter input x(n+1) and the filter coefficient a0 are transferred from a memory (not shown) over the buses and . The delay element produces the data x(n) held therein and then holds the data x(n+1) supplied over the bus .
49
41
42
50
46
42
The multiplier multiplies the data x(n+1) supplied over the bus and the data a0 supplied over the bus to produce the product of a0 and x(n+1) as a multiplication result. The multiplier multiplies the output x(n) of the delay element and the data a0 supplied over the bus to produce the product of a0 and x(n) as a multiplication result.
53
54
51
52
53
54
53
54
These multiplication results are added to the values held by the accumulators and . More specifically, the adder-subtracters and add the multiplication results to the values held by the accumulators and , respectively. Thus the accumulators and hold y(n+1) and y(n), respectively. This completes the operation to implement the FIR filter.
Next, a product sum operation with complex numbers is described. The product sum operation with the complex numbers can be given by the following equation (3):
&lt;PTEXT&gt;&lt;PDAT&gt;(A&plus;jB)&middot;(C&plus;jD)&plus;(E&plus;jF)&equals;(A&middot;C&minus;B&middot;D&plus;E)&plus;j(A&middot;D&plus;B&middot;C&plus;F)&emsp;&emsp;(3)&lt;/PDAT&gt;&lt;/PTEXT&gt;
3
In the equation (), the symbols A through F represent data.
53
54
It is assumed that data E of the complex numbers (E+jF) are stored in the accumulator while the data F are stored in the accumulator . Under this circumstances, the equation (3) is calculated through the following three steps.
41
42
43
46
49
50
41
42
51
52
53
54
The first step involves in the following processing as a preparation for the actual calculation. The data A and the value 0 are transferred from the memory through the buses and . Both the delay elements and hold the data A. Both the multipliers and multiply the data A supplied over the bus with the value 0 supplied over the bus to produce the product of A and 0, which is equal to zero, as a multiplication result. The multiplication result of zero is supplied to the adder-subtracters and where it is added to the value held by the accumulators and .
41
42
43
46
43
49
43
42
50
41
42
51
52
53
54
53
54
In the second step, data B and data C are transferred from the memory through the buses and , respectively. As a result, the delay element holds the data B and the delay element holds the output data A from the delay element . The multiplier multiplies the output data A from the delay element with the data C supplied over the bus to produce the product of A and C as a multiplication result. The multiplier multiplies the data B supplied over the bus with the data C supplied over the bus to produce the product of B and C as a multiplication result. The adder-subtracters and add these multiplication results to the values held by the accumulators and , respectively. Thus, the accumulators and hold (E+A·C) and (F+B·C), respectively.
42
49
43
42
50
46
42
In the third step, the data D is transferred from the memory through the bus . The multiplier multiplies the output data B from the delay element with the data D supplied over the bus to produce the product of B and D as a multiplication result. The multiplier multiplies the output data A supplied from the delay element and the data D supplied over the bus to produce the product of A and D as a multiplication result.
51
53
52
54
53
54
The multiplication result B·D is subtracted in the adder-subtracter from the value held by the accumulator . The multiplication result A·D is added in the adder-subtracter to the value held by the accumulator . Thus the accumulators and hold real and imaginary parts, respectively, of the result of the product sum operation.
As described above, the conventional product sum operation device expedites the product sum operation with the real and the complex numbers by means of the two parallel processing of multiplication and addition. However, the increase in speed to this extent may not be sufficient to provide a satisfactory performance with a possible larger capacity required for communications in the future. Furthermore, the product sum operation device disclosed has a rather complex configuration of an input section including two delay elements and two multiplexers.
FIG. 2
FIG. 2
101
102
103
104
105
107
108
109
110
111
112
113
114
115
115
117
118
105
106
107
110
111
112
107
110
113
114
111
112
115
116
Referring now to , a product sum operation device according to a first embodiment of the present invention is described. In , the product sum operation device according to this embodiment comprises buses , , , and for four words to transfer simultaneously data corresponding to four words from a memory (not shown). The product sum operation device further comprises a latch circuit , multipliers , , and , adder-subtracters and , adders and , accumulators and , and buses and . The latch circuit is for passing through and holding input data in accordance with a control signal . The multipliers through carry out a multiplication operation with the input data to produce a product as a multiplication result. The adder-subtracters and carry out an addition-subtraction operation with the multiplication results obtained by the multipliers through . The adders and add the results obtained by the adder-subtracters and to values held by accumulators and , respectively.
105
102
106
105
102
106
The latch circuit outputs the data as it is that is transferred over the bus when the control signal indicates “ON”, namely, has a high level. The latch circuit temporarily holds the data transferred over the bus when the control signal indicates “OFF”, namely, has a low level and outputs it in the subsequent step.
107
101
103
108
102
104
109
103
105
110
101
104
The multiplier carries out a multiplication operation with the data transferred from the memory over the buses and . The multiplier carries out a multiplication operation with the data transferred from the memory over the buses and . The multiplier carries out a multiplication operation with the data transferred from the memory over the buses and the data supplied from the latch circuit . The multiplier carries out a multiplication operation with the data transferred from the memory over the buses and .
111
107
108
112
109
110
113
111
115
114
112
116
The adder-subtracter adds the multiplication result obtained by the multiplier to or subtracts it from the multiplication result obtained by the multiplier . The adder-subtracter adds the multiplication result obtained by the multiplier to or subtracts it from the multiplication result obtained by the multiplier . The adder adds the result obtained by the adder-subtracter to the value held by the accumulator to produce a sum as an addition result. The adder adds the result obtained by the adder-subtracter to the value held by the accumulator to produce a sum as an addition result.
115
113
116
114
The accumulator holds the addition result obtained by the adder and outputs the held value in the subsequent step. The accumulator holds the addition result obtained by the adder and outputs the held value in the subsequent step.
101
104
107
110
105
107
110
107
101
103
108
102
104
109
103
105
110
101
104
Next, how the product sum operation device according to the present invention provides its function is described. The data corresponding to four words are transferred from the memory over the buses through to the multipliers through and the latch circuit . These data are used for the multiplication operation carried out by the multipliers through . More specifically, the multiplier multiplies the data transferred over the bus with the data transferred over the bus . The multiplier multiplies the data transferred over the bus with the data transferred over the bus . The multiplier multiplies the data transferred over the bus with the data supplied from the latch circuit . The multiplier multiplies the data transferred over the bus with the data transferred over the bus .
107
110
111
112
111
107
108
112
109
110
After these multiplication operations, the products obtained by the multipliers through as multiplication results are subjected to an addition or subtraction operation in the adder-subtracters and . More specifically, the adder-subtracter adds the multiplication result obtained by the multiplier to or subtract it from the multiplication result obtained by the multiplier . The adder-subtracter adds the multiplication result obtained by the multiplier to or subtract it from the multiplication result obtained by the multiplier .
111
112
113
114
115
116
113
111
115
114
112
116
113
114
115
116
The results obtained by the adder-subtracters and are supplied to the adders and where they are added to the values held by the accumulators and . More specifically, the adder adds the result obtained by the adder-subtracter to the value held by the accumulator . The adder adds the result obtained by the adder-subtracter to the value held by the accumulator . The results obtained by the adders and are held by the accumulators and , respectively. This completes one cycle of the step of the product sum operation device according to the present invention.
115
116
113
114
117
118
The values held by the accumulators and obtained in the above-mentioned step are supplied to the adders and , respectively, or supplied to and stored in the memory over the buses and , respectively, in the subsequent step.
Operation is described more in detail for exemplified cases where the product sum operation device is used for the implementation of an FIR filter and for the product sum operation with complex numbers.
First, implementation of the FIR filter is described for the case of a fourth order filter. Like the conventional product sum operation device, a filter output y(n) at a time instant n and a filter output y(n+1) at a time instant (n+1) are calculated in parallel. The calculations according to the equations (1) and (2) are carried out by the product sum operation device according to this embodiment through the following three steps. The control signal should indicate “OFF” during these three steps.
101
102
103
104
105
109
102
The first step involves in the following processing as a preparation for the actual calculation. First, a filter input x(n−4) is transferred from the memory through the bus while a filter input x(n−3) is transferred from the memory through the bus . Simultaneously, the value 0 is transferred from the memory through the buses and . The latch circuit outputs the value held therein (which has no significant relation with the actual calculation) as an input to the multiplier and then hold the filter input x(n−3) transferred over the bus .
107
108
109
105
110
The multiplier multiplies the filter input x(n−4) with the value 0 to produce the product 0 as a multiplication result. The multiplier multiplies the filter input x(n−3) with the value 0 to produce the product 0 as a multiplication result. The multiplier multiplies the output of the latch circuit with the value 0 to produce the product 0 as a multiplication result. The multiplier multiplies the filter input x(n−4) with the value 0 to produce the product 0 as a multiplication result.
111
107
108
112
109
110
115
116
Subsequently, the adder-subtracter adds the multiplication result obtained by the multiplier to the multiplication result obtained by the multiplier to produce the sum 0 as an addition result. On the other hand, the adder-subtracter adds the multiplication result obtained by the multiplier to the multiplication result obtained by the multiplier to produce the sum 0 as an addition result. These addition results (=0) are stored in the accumulators and .
101
102
103
104
105
109
102
In the second step, a filter input x(n−2) is transferred from the memory through the bus and a filter input x(n−1) is transferred from the memory through the bus . Simultaneously, a filter coefficient a3 is transferred from the memory through the bus and a filter coefficient a2 is transferred from the memory through the bus . The latch circuit outputs the value x(n−3) held therein in the first step as an input to the multiplier and then hold the filter input x(n−1) transferred over the bus .
107
108
109
105
110
The multiplier multiplies the filter input x(n−2) with the filter coefficient a3 to produce the product of a3 and x(n−2), i.e., a3·x(n−2) as a multiplication result. The multiplier multiplies the filter input x(n−1) with the filter coefficient a2 to produce the product of a2 and x(n−1), i.e., a2·x(n−1) as a multiplication result. The multiplier multiplies the filter input x(n−3) held by the latch circuit in the first step with the filter coefficient a3 to produce the product of a3 and x(n−3), i.e., a3·x(n−3) as a multiplication result. The multiplier multiplies the filter input x(n−2) with the filter coefficient a2 to produce the product of a2 and
x(n−2), i.e., a2·x(n−2) as a multiplication result.
111
107
108
112
109
110
113
114
115
116
115
116
Then, the adder-subtracter adds the multiplication result obtained by the multiplier to the multiplication result obtained by the multiplier to produce the sum of a2 multiplied by x(n−1) and a3 multiplied by x(n−2), i.e., (a2·x(n−1)+a3x·(n−2) as an addition result. On the other hand, the adder-subtracter adds the multiplication result obtained by the multiplier to the multiplication result obtained by the multiplier to produce the sum of a2 multiplied by x(n−2) and a3 multiplied by x(n−3), i.e., (a2·x(n−2)+a3·x(n−3) as an addition result. These addition results are added by the adders and to the values of 0 stored in the accumulators and , respectively. Thus, the accumulator stores a2·x(n−1)+a3·x(n−2) while the accumulator stores a2·x(n−2)+a3·x(n−3).
101
102
103
104
105
109
102
In the third step, a filter input x(n) is transferred from the memory through the bus and a filter input x(n+1) is transferred from the memory through the bus . Simultaneously, a filter coefficient al is transferred from the memory through the bus and a filter coefficient a0 is transferred from the memory through the bus . The latch circuit outputs the value x(n−1) held therein in the second step as an input to the multiplier and then hold the filter input x(n+1) transferred over the bus .
107
108
109
105
110
The multiplier multiplies the filter input x(n) with the filter coefficient a1 to produce the product of a1 and x(n), i.e., a1·x(n) as a multiplication result. The multiplier multiplies the filter input x(n+1) with the filter coefficient a0 to produce the product of a0 and x(n+1), i.e., a0·x(n+1) as a multiplication result. The multiplier multiplies the filter input x(n−1) held by the latch circuit in the second step with the filter coefficient a1 to produce the product of a1 and x(n−1), i.e., a1·x(n−1) as a multiplication result. The multiplier multiplies the filter input x(n) with the filter coefficient a0 to produce the product of a0 and x(n), i.e., a0·x(n) as a multiplication result.
111
107
108
112
109
110
113
114
115
116
115
116
Subsequently, the adder-subtracter adds the multiplication result obtained by the multiplier to the multiplication result obtained by the multiplier to produce the sum of a0 multiplied by x(n+1) and a1 multiplied by x(n), i.e., (a0·x(n+1)+a1·x(n) as an addition result. On the other hand, the adder-subtracter adds the multiplication result obtained by the multiplier to the multiplication result obtained by the multiplier to produce the sum of a0 multiplied by x(n) and a1 multiplied by x(n−1), i.e., (a0·x(n)+a1·x(n−1) as an addition result. These addition results are added by the adders and to the values stored in the accumulators and , respectively. Thus, the accumulator holds y(n±1) while the accumulator holds y(n).
Though the above-mentioned embodiment is for a case of the fourth order FIR filter, it is apparent that the present invention is applicable to the FIR filter having any number of the order. It is also apparent that the present invention is applicable to a convolution operation in an IIR filter rather than an FIR filter.
The above-mentioned embodiment requires three steps for the calculation with the equations (1) and (2), which means 40% reduction of the steps as compared with the five-step calculation carried out by the conventional product sum operation device. The higher the order of the filter is, the closer to 50% the percentage of the reduction in steps becomes gradually.
115
116
Next, a product sum operation with complex numbers given by the equation (3) is described. The data E of the complex numbers (E+jF) are stored in the accumulator while the data F are stored in the accumulator . The control signal indicates “ON”.
107
108
109
110
101
102
103
104
Under such a circumstances, the equation (3) is calculated according to the following single step. First, the data A, B, C, and D are transferred from the memory to the multipliers , , , and through the buses , , , and , respectively.
107
108
109
110
The multiplier multiplies the data A with the data C to produce the product of A and C, i.e., A·C. The multiplier multiplies the data B with the data D to produce the product of B and D, i.e., B·D. The multiplier multiplies the data B with the data C to produce the product of B and C, i.e., B·C. The multiplier multiplies the data A with the data D to produce the product of A and D, i.e., A·D.
111
108
107
111
109
110
Subsequently, the adder-subtracter subtracts the product of B and D obtained by the multiplier from the product of A and C obtained by the multiplier to produce a subtraction result of (A·C−B·D). The adder-subtracter adds the product of B and C obtained by the multiplier to the product of A and D obtained by the multiplier to produce an addition result of (A·D+B·C).
113
114
115
116
115
116
These results are supplied to the adders and where they are added to the values held by the accumulators and , respectively. Thus the accumulator holds the value equal to A multiplied by C minus B multiplied by D plus E, that is, (A·C−B·D+E) and the accumulator holds the value equal to A multiplied by D plus B multiplied by C plus F, that is (A·D+B·C+F).
The conventional product sum operation device requires three steps for the product sum operation with the complex numbers. On the contrary, the product sum operation device according to the present invention requires only single step for the equivalent calculation. This means an approximately 67% reduction of the steps.
FIG. 3
FIG. 3
201
202
203
204
107
110
107
201
108
202
109
203
110
204
Referring to , a product sum operation device according to a second embodiment of the present invention is described. In , the product sum operation device according to this embodiment is different from the first embodiment in the connections of buses , , , and which are used for transferring data from a memory (not shown) to the multipliers through . The multiplier is supplied with the data transferred from the memory through the bus as a first input and a second input for the multiplier. The multiplier is supplied with the data transferred from the memory through the bus as a first input and a second input for the multiplier. The multiplier is supplied with the data transferred from the memory through the bus as a first input and a second input for the multiplier. The multiplier is supplied with the data transferred from the memory through the bus as a first input and a second input for the multiplier.
Now, how the product sum operation device according to the present invention provides its function is described. As an example, description is made for a parallel calculation of a square sum according to the following equations (4) and (5):
&lt;PTEXT&gt;&lt;PDAT&gt;P&equals;A&lt;/PDAT&gt;&lt;HIL&gt;&lt;SP&gt;&lt;PDAT&gt;2&lt;/PDAT&gt;&lt;/SP&gt;&lt;/HIL&gt;&lt;PDAT&gt;&plus;B&lt;/PDAT&gt;&lt;HIL&gt;&lt;SP&gt;&lt;PDAT&gt;2&lt;/PDAT&gt;&lt;/SP&gt;&lt;/HIL&gt;&lt;PDAT&gt;&plus;C&lt;/PDAT&gt;&lt;HIL&gt;&lt;SP&gt;&lt;PDAT&gt;2&lt;/PDAT&gt;&lt;/SP&gt;&lt;/HIL&gt;&lt;PDAT&gt;&plus;D&lt;/PDAT&gt;&lt;HIL&gt;&lt;SP&gt;&lt;PDAT&gt;2&lt;/PDAT&gt;&lt;/SP&gt;&lt;/HIL&gt;&lt;PDAT&gt;&emsp;&emsp;(4)&lt;/PDAT&gt;&lt;/PTEXT&gt;
&lt;PTEXT&gt;&lt;PDAT&gt;Q&equals;E&lt;/PDAT&gt;&lt;HIL&gt;&lt;SP&gt;&lt;PDAT&gt;2&lt;/PDAT&gt;&lt;/SP&gt;&lt;/HIL&gt;&lt;PDAT&gt;&plus;F&lt;/PDAT&gt;&lt;HIL&gt;&lt;SP&gt;&lt;PDAT&gt;2&lt;/PDAT&gt;&lt;/SP&gt;&lt;/HIL&gt;&lt;PDAT&gt;&plus;G&lt;/PDAT&gt;&lt;HIL&gt;&lt;SP&gt;&lt;PDAT&gt;2&lt;/PDAT&gt;&lt;/SP&gt;&lt;/HIL&gt;&lt;PDAT&gt;&plus;H&lt;/PDAT&gt;&lt;HIL&gt;&lt;SP&gt;&lt;PDAT&gt;2&lt;/PDAT&gt;&lt;/SP&gt;&lt;/HIL&gt;&lt;PDAT&gt;&emsp;&emsp;(5)&lt;/PDAT&gt;&lt;/PTEXT&gt;
wherein A through H represent data.
107
110
201
204
107
110
The calculations with the equations (4) and (5) requires three steps. First step involves in the following processing as a preparation for the actual calculation. First, the value 0 is transferred from the memory to the multipliers through via the buses through . The multipliers through calculates the square of zero and produces the result of zero as a multiplication result.
111
107
108
112
109
110
115
116
Subsequently, the adder-subtracter sums the multiplication results obtained by the multipliers and and produces the result of zero as an addition result. The adder-subtracter sums the multiplication results obtained by the multipliers and and produces the result of zero as an addition result. These addition results are stored in the accumulators and , respectively.
107
108
109
110
201
202
203
204
107
108
109
110
2
2
2
2
In the second step, the data A, B, E, and F are transferred from the memory to the multipliers , , , and through the buses , , , and , respectively. The multiplier calculates the square of the data A and produces Aas a multiplication result. The multiplier calculates the square of the data B and produces Bas a multiplication result. The multiplier calculates the square of the data E and produces Eas a multiplication result. The multiplier calculates the square of the data F and produces Fas a multiplication result.
111
107
108
112
109
110
113
11
4
115
116
115
116
2
2
2
2
2
2
2
2
Next, the adder-subtracter sums the multiplication results obtained by the multipliers and and produces the result of (A+B) as an addition result. Likewise, the adder-subtracter sums the multiplication results obtained by the multipliers and and produces the result of (E+F) as an addition result. These addition results are supplied to the adders and where they are added to the value of 0 held by the accumulators and , respectively. Thus, the accumulator holds (A+B) while the accumulator holds (E+F).
107
108
109
110
201
202
203
204
In the third step, the data C, D, G, and H are transferred from the memory to the multipliers , , , and through the buses , , , and , respectively.
107
108
109
110
2
2
2
2
The multiplier calculates the square of the data C and produces Cas a multiplication result. The multiplier calculates the square of the data D and produces Das a multiplication result. The multiplier calculates the square of the data G and produces Gas a multiplication result. The multiplier calculates the square of the data H and produces Has a multiplication result.
111
107
108
112
109
110
113
114
115
116
115
116
2
2
2
2
Then, the adder-subtracter sums the multiplication results obtained by the multipliers and and produces the result of (C+D) as an addition result. Likewise, the adder-subtracter sums the multiplication results obtained by the multipliers and and produces the result of (G+H) as an addition result. These addition results are supplied to the adders and where they are added to the values held by the accumulators and , respectively. Thus, the accumulator holds the result P given by the equation (4) while the accumulator holds the result Q given by the equation (5).
FIG. 4
FIG. 4
FIGS. 2 and 3
Now, referring to , a product sum operation device according to a third embodiment of the present invention is described. In , the product sum operation device according to this embodiment is a modification of the embodiments shown in configured to achieve these two embodiments as a single embodiment.
301
302
303
304
107
108
109
110
305
306
307
308
301
302
303
304
309
309
The product sum operation device according to this embodiment is different from the first embodiment in the connections of buses , , , and which are used for transferring data from a memory (not shown) to the multipliers , , , and , respectively. Furthermore, other differences are multiplexers , , and placed on the buses , , and and the use of a selection signal . The selection signal is a signal for functioning the operation device as the product sum operation device according to the first embodiment and/or as the product sum operation device according to the second embodiment.
309
309
305
309
301
303
305
303
309
305
301
309
306
309
302
304
306
304
309
306
302
309
The selection signal indicates “OFF” when the operation device is intended to be used as the product sum operation device according to the first embodiment. The selection signal indicates “ON” when the operation device is intended to be used as the product sum operation device according to the second embodiment. The multiplexer is supplied with the selection signal and the data transferred from the memory through the buses and . The multiplexer outputs the data supplied over the bus when the selection signal indicates “OFF”. The multiplexer outputs the data supplied over the bus when the selection signal indicates “ON”. The multiplexer is supplied with the selection signal and the data transferred from the memory through the buses and . The multiplexer outputs the data supplied over the bus when the selection signal indicates “OFF”. The multiplexer outputs the data supplied over the bus when the selection signal indicates “ON ”.
307
309
303
105
307
105
309
307
303
309
308
309
301
304
308
301
309
308
304
309
The multiplexer is supplied with the selection signal , the data transferred through the bus and the data supplied from the latch circuit . The multiplexer outputs the data supplied from the latch circuit when the selection signal indicates “OFF”. The multiplexer outputs the data supplied over the bus when the selection signal indicates “ON”. The multiplexer is supplied with the selection signal and the data transferred from the memory through the buses and . The multiplexer outputs the data supplied over the bus when the selection signal indicates “OFF”. The multiplexer outputs the data supplied over the bus when the selection signal indicates “ON”.
107
301
305
108
302
306
109
303
307
110
304
308
The multiplier multiplies the data transferred from the memory through the bus with the data supplied from the multiplexer to produce the product as a multiplication result. The multiplier multiplies the data transferred from the memory through the bus with the data supplied from the multiplexer to produce the product as a multiplication result. The multiplier multiplies the data transferred from the memory through the bus with the data supplied from the multiplexer to produce the product as a multiplication result. The multiplier multiplies the data transferred from the memory through the bus with the data supplied from the multiplexer to produce the product as a multiplication result.
309
309
From the viewpoint how the operation device provides its functions, the operation device of this embodiment is similar to the one described in conjunction with the first embodiment when the selection signal indicates “OFF” and is similar to the one described in conjunction with the second embodiment when the selection signal indicates “ON”.
The above-mentioned three embodiments may be applied as an operation device for digital signal processors and microprocessors.
As described above, the product sum operation device according to the present invention has more multipliers and adder-subtracters than the conventional product sum operation devices do. This contributes to the acceleration of product sum operations as compared with the conventional product sum operation devices. In addition, the product sum operation device implemented by the first and the second embodiments requires no multiplexer that is required in the conventional product sum operation device. This simplifies the configuration and structure of the operation unit.
The product sum operation device according to the present invention has a switch for use in controlling the supply of the data. More specifically, the control signal is used to control whether the data transferred through the data transfer bus is immediately supplied to the multiplier or is supplied later when the subsequent data is transferred over the data transfer bus. Therefore, the product sum operation device of the present invention is suitable for various product sum operations such as a product sum operation with complex numbers and product sum operation with real numbers by means of convolution operation for providing an FIR filter.
Furthermore, the product sum operation device according to the present invention has the multipliers capable of calculating a square of the data transferred over the data transfer bus. This means that the product sum operation device can be used for a fast calculation of a square sum.
In addition, the product sum operation device according to the third embodiment has the multiplexers that are adapted to supply to the first through the fourth multipliers either the data transferred through the first through the fourth data transfer buses or the data sent through another bus. Accordingly, various product sum operation including the square sum operation can be carried out at a high speed. | |
Updated: May 2, 2022
Anyone who is in the process of navigating a divorce hopes that the process will go as quickly and smoothly as possible. Unfortunately, many people make significant financial mistakes as they go through the divorce process. The financial decisions that a person makes during divorce can greatly contribute to his or her economic situation afterward, which is why it is important to have an attorney on your side helping you look out for your financial best interests. The following will review some tips to consider when making common financial decisions that arise during a divorce.
HAVE AN ACCURATE PICTURE OF YOUR FINANCES
After filing for a divorce, one of the best first steps to take is to assess your finances as well as to establish a precise budget. It is also important to understand that your financial situation will likely change during and after your divorce, so periodically review your finances over the next few years.
REMEMBER SHARED DEBTS
During the division of assets phase that occurs in a divorce, a court of law will also divide your liabilities and debts. To prepare for this phase, it is critical to have a firm idea of your shared debts, which might include things like car loans, private loans, or credit card debt.
CONSIDER TAX ISSUES
During the divorce process, it is critical to understand that the dissolution of your marriage will result in a number of tax changes. After all of your assets and liabilities have been divided, you might discover that you owe additional money in taxes. To avoid the unexpected come tax season, remain mindful of the tax implications of your divorce and discuss them with your attorney and accountant.
DO NOT HIDE ASSETS
Although it can be tempting to hide assets during a divorce so that they do not end up divided between you and your spouse, this type of behavior is illegal. To avoid negative consequences, it is better to honestly disclose details about all of your assets. Rather than hide your assets, you should discuss your agenda with your attorney and then develop a strategy to increase your chances of obtaining these assets during the property division process.
CLOSE SHARED ACCOUNTS
You should not continue sharing finances with your spouse after a divorce. If your spouse is still able to access your financial accounts, there is a risk that your spouse might attempt to monitor your spending or make unauthorized withdrawals.
SPEAK WITH A EXPERIENCED DIVORCE LAWYER
Even though there are a number of financial mistakes that can be made during divorce, by following the suggestions listed above, you can decrease the chances that you end up facing financial difficulties. If you are currently navigating the divorce process, it is also often a wise idea to obtain the assistance of an experienced divorce attorney. Contact a divorce lawyer in Glendale at the Monahan Law Firm PLC today to schedule an initial case evaluation. | https://pjmfirm.com/critical-financial-considerations-when-navigating-a-divorce/ |
USA’s Madison Hubbell and Zach Donohue maintained their overnight lead to win the gold after the Free Dance at 2018 Skate America. Italy’s Charlene Guignard and Marco Fabbri also held their position, earning the silver, while Russia’s Tiffani Zahorski and Jonathan Guerreiro captured bronze.
Hubbell and Donohue gave a passionate performance to modern music from the Romeo and Juliet soundtrack, showing soft knees and expression throughout. It’s obvious that major changes have been made to the routine since they performed it at 2018 U.S. International Classic where they won gold. The U.S. champions earned no less than +3 positive Grades of Execution (GOE) on all elements, and in fact, earned mostly +4 for their twizzles. The team also earned 122.39 for the Free Skate and broke the 200-mark with 200.82 overall.
“Zach and I put in a lot of work since our first competition in Salt Lake to really dive into the emotions of Romeo and Juliet and to put more skating quality, more power into the program,” said Hubbell. “I think we were able to show those changes well, but we’re also looking forward to more and more improvement, and hopefully, have a cleaner skate next week in Skate Canada.”
Guignard and Fabbri gave a technically solid and expressive performance in their delightful routine to music from the La La Land soundtrack. The two-time Olympians also showed strong level 4 twizzles as well as difficult level 4 lifts. They scored 117.29 for the Free Dance and maintained second place overall with 192.30 points.
“We felt a lot better than last night,” said Fabbri. “Even though the legs were more tired, but we probably were a bit more focused. It (the first Grand Prix medal) is a great reward for us. We started from zero when Charlene and I started skating together. I had just started ice dance and she didn’t have international experience. We are proud of what we have achieved.”
Zagorsiki and Guerreiro showed attitude in their complete new style to the music of “Blues” for Klook by Eddie Louis, but received deductions for extended lifts. Although the 2018 Russian national bronze medalists haven’t had much training time, they placed a respectable fourth in the free skate with 108.08, maintaining third overall (181.38).
Click here to read more. | http://whytehousereport.com/2018/10/24/hubbell-and-donohue-win-tenth-skate-america-medal-for-a-good-start-to-season/ |
Sydney’s Westlink M7 Motorway opened to traffic in December 2005 (six months ahead of schedule):
- The motorway runs for 40 km (25 miles), and connects three other major motorways in Sydney (M2, M4, M5); it is situated within a high growth industrial and residential corridor
- The motorway includes 17 access points, 38 over-passes and underpasses, 144 bridges and a 40 km uninterrupted cycling and walking path.
The motorway’s route runs within a narrow industrial and residential corridor experiencing strong growth (almost 5,000 residential dwellings lie in close proximity to the route). Accordingly, the mitigation of noise and vibration (during both the construction and operational phases of the project) was identified early in the design process as one of the project’s key environmental factors.
Ultimately, SLR’s noise and vibration consultancy, which spanned the full duration of the project, included the design and optimisation of 160,000 m2 of noise walls and over 1,200 property treatments.
SLR undertook the following works:
- Tender Phase: Tender noise mitigation costings (noise walls, low noise pavement and property treatments)
- Design Phase: Baseline noise surveys; operational noise predictions; attenuation recommendations covering pavement (low-noise options), noise walls and at-residence insulation; maximum noise level assessment (Compression Brake Noise Minimisation Strategy); Construction Noise and Vibration Management Plan
- Stakeholder Engagement: Community consultation (pre and during construction)
- Construction Phase: Construction noise and vibration monitoring
- Post-Opening Phase: Post-compliance (operational) noise monitoring and residual property treatments. | https://slrconsulting.com/en/projects/westlink-m7-motorway |
Whilst boarding the plane to head back home to Australia, I was overcome with so many conflicting mixed emotions. Deep sadness because it was over, pure happiness because it was amazing, excitement for the opportunities it may have given us, and a strong sense of disbelief. We had just been, and showcased at New York Fashion Week. Even saying it now, I still don’t feel it’s sunk in.
~
We landed in New York on a balmy Tuesday afternoon, and hopped straight into a big yellow taxi (so cliché, I know) which quickly flowed into the peak hour traffic and hustle and bustle of the city. After two hours in traffic we arrived at our cute little Air BnB in West Village, a beautiful and homely neighbourhood, with bright green floating trees sweeping each sidewalk. Next door boasted a relaxing yet vibrant café, Grounded Organic Coffee and Tea House, which became our second home whilst in the states. We ventured out and wandered the neighbourhood as the sun went down, pinching ourselves whilst we explored. After visiting a local market and stocking up on the necessities - muffins and coffee – we became overcome with exhaustion from the travel, headed back and nestled in for our first night away from home.
The next day we were up early, eager to start exploring. We took the Subway and popped up in lively Times Square. We did a little shopping, just a little, and sussed out some backdrops and locations for a photoshoot, which we were to do here in just over a weeks time. Before long, the day was done.
The coming days quickly went, and Ashlee proved she was a pure machine. I have watched her work hard, but when it came to New York she never lost sight of the goal and worked harder than ever. She smashed out more elements of her new website, gathered loads of social media content, arranged the photoshoot including a NYC model and hair and makeup artists, she fixed up last minute details of the crowns, responded to emails and enquiries back home and even organised collaborations.
This woman worked day and night on our trip, she did not stop, but after a few days of a hefty workload, it was time for us to focus on the models fittings day! We were taken to a large multi-storey building not far from our apartment in West Village. From the foyer the building looked upper-class and sophisticated, however upon getting out at our designated floor, we were greeted with colourful hallways, and swarms of people. Some dancing in the corridor, some practicing lines in the mirror. This was a full-blown audition building, just like in the movies. There were show posters on all the walls, audition times pinned to doors, people lined up ready to head in to have their go. You couldn’t wipe the smiles off our faces in that moment.
We entered our room and met the Fashion Palette team and began pulling out all the product, awaiting the models arrival. We were then given our backstage passes to NYFW, and cue excited laughter/disbelief/happy tears. The models arrived soon after and Ash quickly started fitting them with their headpieces and talking to them about our vision for the show and what she wanted them to do, whilst I fitted them with their lingerie and shirts and Chris handled accessories. We were scribbling down notes, recording shoe sizes and shirt sizes, their allocated pieces, you name it. The day was over before we knew it, and it was now time to prepare for the pinnacle of the trip. Our beautiful photographer Jess Tucker from back home, made the trek over to the states and met us that night in New York ready to share our excitement.
When the day came around for the runway show, we popped a bottle of Moët to celebrate beforehand, and I looked over to Ash, Chris and Jess and we all sat in silence and really stopped to take it all in. After a few minutes we joked again for the 36th time about how surreal it was that we were here, had couple more glasses of champas to ease the nerves and it was time to go in. It was time for the Ashlee Lauren New York Fashion Week Runway show.
Backstage was buzzing. There were hair and makeup stations with the dressing-room styled mirrors, clothing racks allocated to each model filled with brightly coloured and beautifully detailed garments, camera-men and journalists frolicking around madly trying to interview the designers, models running around naked frantically trying to find their racks and next outfit and organisers with their clip-boards trying to keep everything some-what running smoothly. Then there was me and Ash, I feel like deer in headlights, standing still as everything swung around us trying to take it all in.
Ash was pulled aside for interview after interview, and I spotted a model I recognised from fittings looking for her station. I headed over and started dressing her, and before long more models were ready. We had half our models dressed, fitted with headpieces ready for the show. We were still waiting on 5 more models, and time to the show was dwindling away. Jess, from Fashion Palette started heading our way, sensing our confusion as to where the other models were. Then she broke the news, the agency had double-booked the models, so five of them would be a no-show.
To be continued.... | https://aeleste.com/blogs/ashlee-lauren-my-story/new-york-fashion-week-part-1 |
Small Steps Save Lives: Awareness Campaign on COVID-19 in Palestine
Consequently, schools and universities closed impacting 1.3 million students, who join an estimated 1.4 billion learners locked out of school around the world, according to UNESCO. While students and alumni of the Al Fakhoora Scholarship and Empowerment Programme usually work with children and adolescents in many schools throughout Gaza, developing their leadership skills, the shutdown of schools reflected a unique challenge. Like so many young people around the world right now, these inspirational leaders were not deterred.
Equipped with hygiene and educational materials and their passion to engage with the community, groups of Al Fakhoora students from different universities with medical specialities developed a plan to deploy throughout Gaza. Their goal was simple: help the community learn the best preventative techniques to avoid spreading COVID-19. They engaged with the public in markets, parks, beach cafeterias and shops, as well as connecting with passers-by in the Gaza Strip. They delivered short talks aimed at advising individuals on how to maintain ‘social distancing’ by keeping space between each other; best hand washing techniques and practices; how to politely avoid shaking hands and other cultural practices like hugs; and generally communicating about staying away from gatherings to minimize the possibility of spreading the COVID-19 virus.
"We have assembled, proactively, to support our community in Gaza to avoid the Corona outbreak. There is no place for fear now, we cannot just sit by terrified waiting for Corona to hit Gaza. Along with my Al Fakhoora Fellows in medicine, nursing and paramedics, we have launched a campaign to raise awareness in our community about the preventative techniques that can be used to avoid COVID-19 infection and spread. Although our efforts may seem limited, we do believe that more hands are needed to save our future," said Ahmad Al Sahar, 6th year medical student, Islamic University, Gaza.
Nadeen Shaath, Al Fakhoora Alumni specializing in medical laboratories, described the initiative as a "…social responsibility to spread awareness of the risks of coronavirus (COVID-19) among the people of Gaza and beyond. I have started to record short videos, giving practical guidelines regarding the ‘right practices’ of washing hands, which could be even more effective than using anti-bacterial gel. This is an easy thing can be done regularly to avoid infection after touching any surface. It's not about buying expensive materials, but about changing the daily culture of hygiene…"
Actively guiding children, teaching them with simple words, Abduallah AbuNamoos, a nursing student, stated, "…targeting children is at the heart of our campaign. Working with children is very sensitive. It is about good communication, rather than simplifying the story about the Corona outbreak. With the information we provide, we aim to decrease fears and promote positivity and hope. COVID-19 is a virus that everyone needs to avoid, not only because it can be fatal, but also due to its serious impact on health services. Our approach is to encourage taking hygiene practices more seriously."
Sami Naem, a fifth-year medical student who actively organized the campaign, concluded that, "…this comes as a part of our effort to give back to our communities. As Al Fakhoora students we know we must do more while staying as safe as possible. Even while at home, we will utilize social platforms to spread awareness and fight the virus. I find it a blessing to be going through my educational journey as an Al Fakhoora student. Being empowered with knowledge and skills allows us to contribute and help people in need. I wish I could contribute to the research to find a solution to this. Still, as Al Fakhoora students, we will continue to give as best as we can." | https://www.educationaboveall.org/media-centre/news/small-steps-save-lives-awareness-campaign-covid-19-palestine |
Acetylcholinesterase inhibitors (e.g., pyridostigmine bromide) are used for neuromuscular blockade (NMB) reversal in patients undergoing surgery under general anesthesia (GA). Concurrent use of anticholinergic agents (e.g., glycopyrrolate) decreases cholinergic side effects but can impede bowel movements. Sugammadex has no cholinergic effects; its use modifies recovery of gastrointestinal (GI) motility following laparoscopic cholecystectomy compared to pyridostigmine/glycopyrrolate. This study evaluated the contribution of sugammadex to the recovery of GI motility compared with pyridostigmine and glycopyrrolate.
Methods
We conducted a prospective study of patients who underwent laparoscopic cholecystectomy. Patients were randomly allocated to the experimental group (sugammadex, Group S) or control group (pyridostigmine-glycopyrrolate, Group P). After anesthesia (propofol and rocuronium, and 2% sevoflurane), recovery was induced by injection of sugammadex or a pyridostigmine-glycopyrrolate mixture. As a primary outcome, patients recorded the time of their first passage of flatus (‘gas-out time’) and defecation. The secondary outcome was stool types.
Results
One-hundred and two patients participated (Group S [n = 49], Group P [n = 53]). Mean time from injection of NMB reversal agents to gas-out time was 15.03 (6.36–20.25) h in Group S and 20.85 (16.34–25.86) h in Group P (P = 0.001). Inter-group differences were significant. Time until the first defecation as well as types of stools was not significantly different.
Conclusions
Sugammadex after laparoscopic cholecystectomy under GA resulted in an earlier first postoperative passage of flatus compared with the use of a mixture of pyridostigmine and glycopyrrolate. These findings suggest that the use of sugammadex has positive effects on the recovery of GI motility.
Introduction
The use of acetylcholinesterases (e.g., rocuronium bromide) is essential in achieving neuromuscular blockade (NMB) for surgery under general anesthesia (GA), which requires a deep NMB . For NMB reversal following the use of acetylcholinesterase, acetylcholinesterase inhibitors (AChEIs)(e.g., neostigmine and pyridostigmine) are used as reversal agents. In addition, anticholinergic agents such as atropine and glycopyrrolate have been used to reduce the resulting cholinergic side effects, which include bradycardia and increased secretions [2,3].
Regarding bowel movements, AChEIs increase motility, whereas anticholinergic agents decrease it. Recovery to normal bowel movements and prevention of postoperative ileus are important for early recovery after surgery. One study reported that neostigmine, an AChEI, decreases postoperative ileus (a type of bowel obstruction) . However, research comparing the effects of drugs with opposing effects on bowel movements has yet to be conducted.
Sugammadex, a recently introduced reversal agent, has no cholinergic side effects, and thus, it does not require the use of anticholinergic agents . A number of studies have confirmed that the use of sugammadex for recovery from anesthesia leads to fewer respiratory complications and less residual NMB compared with the conventionally used AChEIs and anticholinergic agents, and that it contributes to enhanced recovery after surgery (ERAS®) [5,6]. ERAS® addresses the prevention of postoperative ileus as an important issue, for which investigations have been conducted to evaluate various preventive mechanisms, including gum chewing, early enteral nutrition, and laparoscopic surgery. In this context, few studies have been conducted to investigate the effects of sugammadex on bowel movements [7-10]. Moreover, only a small number of studies have compared the recovery of intestinal movement in groups administered AChEIs and anticholinergic agents and those administered sugammadex, although the studies were conducted retrospectively . This study, therefore, aimed to evaluate the contribution of sugammadex as a reversal agent to the recovery of gastrointestinal (GI) motility in patients undergoing laparoscopic cholecystectomy compared to the contribution of the combination of pyridostigmine and glycopyrrolate.
Materials and Methods
This study was conducted with the approval of the Institutional Review Board (IRB) of Daegu Fatima Hospital (IRB approval number: DFH18MRIO366) and this study was registered at https://cris.nih.go.kr (KCT0004330). We explained to the patients the purpose of this prospective study and obtained their written consent before commencing the study.
We explained the method of anesthesia to the patients scheduled for laparoscopic cholecystectomy under GA as well as to their guardians. They were also informed concerning the use of NMB agents and the need for NMB reversal agents. We then explained to them the merits and demerits of the two types of reversal agents and obtained their consent to the randomized allocation of a drug.
Patient characteristics
We selected patients between ages 20 and 70 years who were scheduled for GA-induced laparoscopic cholecystectomy and had American Society of Anesthesiologists (ASA) physical status I or II.
Exclusion criteria
We excluded patients requiring emergency care due to their inability to control nothing by mouth (NPO) fasting time, and those diagnosed with diabetes, ulcerative colitis, or Crohn’s disease, all of which can affect patient GI motility. Patients with renal dysfunction were also excluded [12,13].
Intervention
Fig. 1 shows the flow diagram of this study.
The study participants were allocated randomly to the experimental group, Group S (sugammadex), and the control group, Group P (pyridostigmine). Preoperatively, both groups fasted from midnight on the day of surgery and then consumed two cans of oral carbohydrate solutions (NONPO® 400 ml, Daesang Life Science, Korea) 4 h prior to surgery [8,9,14]. As premedication, midazolam 2 mg (IM) and famotidine 20 mg (IV) were administered 30 min before surgery. Upon arrival at the operation room, the patients were subjected to the induction of GA using propofol 2 mg/kg and rocuronium 0.6 mg/kg, both intravenously, while train-of-four (TOF) monitoring was in progress. Intubation proceeded with the confirmation of a TOF ratio of 0. To maintain GA, we used FiO2 0.5 and 2% sevoflurane (inhalational anesthetic) and injected a mixture of remifentanil 2 mg and normal saline 50 ml via infusion pump. For intraoperative fluid management, we avoided calcium ions, which can induce constipation. Instead, we used crystalloids (plasma solution A) intravenously at rates of 4 cc/kg/h for the first 10 kg, 2 cc/kg/h for the second 10 kg, and 1 cc/kg/h for every kg above 20 kg according to the 4-2-1 rule, and additional 1 cc/kg/h according to perioperative fluid management guidelines [15,16].
Following completion of surgery, administration of sevoflurane was stopped for recovery from GA. For NMB reversal, when a TOF of 2 or above was observed, we intravenously injected the patients with one of the two NMB reversal agents, i.e., sugammadex 2 mg/kg (Group S) or pyridostigmine 0.2 mg/kg and glycopyrrolate 0.008 mg/kg (Group P), and we recorded the time of injections. When the patients’ TOF ratio was confirmed to have reached a minimum of 90%, we proceeded with extubation and transported the patients to the post-anesthesia care unit (PACU). Upon arrival in the PACU, palonosetron 0.075 mg was administered intravenously to prevent postoperative nausea and vomiting (PONV).
For pain control, we intravenously administered a mixture of propacetamol 2 g and normal saline 100 ml; in addition, as patient-controlled analgesia, instructions were provided for administration of normal saline 100 ml mixed with ketorolac tromethamine 240 mg. When patients complained of continued postoperative pain, with numerical rating scale of 6 or above, we provided additional pain control through intravenous administration of fentanyl 1 μg/kg (maximum of two injections). The amount of intraoperative remifentanil used was computed based on the amount of mixed fluid used recorded immediately after surgery.
After the patients were moved to their rooms, we intravenously administered tramadol PRN up to three times when pain intensity of 5 or above was indicated on the visual analogue scale. The patients maintained their NPO fasting throughout the day of the surgery.
Outcome
The patients were instructed to consume carbohydrate drink (NONPO® 400 ml, Daesang Life Science, Korea) on the morning of postoperative day (POD) 1 and to start with soft foods in the afternoon. To evaluate the patients’ bowel movement recovery, following intake of food, they were instructed to record the time of their first passage of flatus (‘gas out’) in their rooms and the time of the first defecation to the minute. As a primary outcome, data on the time elapsed between the injection of NMB reversal agents and the first gas-out and defecation were collected and compared. As a secondary outcome, the presence of any adverse effect (such as nausea, vomiting, and dry mouth), as well as the types of stools additionally based on the Bristol stool scale (Fig. 2), were recorded for comparison.
Randomization
We employed simple randomization with a closed envelope technique for the allocation of the reversal agents. Two sealed envelopes were prepared, each containing a mark for Group S or Group P. Regarding patient assignments, neither the patients nor we were allowed to select or check the envelopes. Third parties with no involvement in the study selected the envelopes and then delivered them to other individuals (‘fourth parties’), who did not participate in the observation of the test results. The fourth parties were those who opened and checked the envelope contents. According to the allocations revealed, each drug (sugammadex vs. pyridostigmine and glycopyrrolate) was prepared to be administered as a reversal agent using 5 cc syringes and normal saline. Prepared in equal amounts, both agents were delivered back to the third parties and then administered randomly to the patients. The drug allocation chart was maintained by the fourth parties until the completion of data collection. It was not until delivery of the analyzed and compared results from the data from the fourth parties that we gained access to the details of the randomization. The patients and third parties were also denied access to the information up to that point.
Sample size
Power analysis was conducted using G*Power 3.1.9.4. Sample size of the previous study was based on the gas-out time in the general surgery ward . Likewise, estimates of effective sizes were made using our previous record of cholecystectomy patients in the general surgery ward. An effect size of 0.527 was calculated using a mean gas-out time of 17 h with a standard deviation (SD) of 7.4 h in the sugammadex group and 20.6 h with a SD of 6.2 h in pyridostigmine group. A sample size of 48 patients per group was found to provide 80% power to detect the effect size with a set α of 0.05 for a two-sided design. A potential drop-out rate of 10% was taken into account. Finally, the study included a total 106 patients who underwent laparoscopic cholecystectomy.
Statistical analysis
We used Student’s t-test to analyze the height, weight, and age of the patients, the amount of remifentanil administered intraoperatively, and the amount of fentanyl administered in the PACU. Sex and ASA physical status classification (ASA scores) of the patients were examined with Fisher’s exact test. The Mann-Whitney U test was used for the analysis of gas-out and defecation times and Fisher’s exact test for stool type and analysis of data on adverse effects.
Results
A total of 106 patients were initially enrolled in the study. Of these, three patients were excluded owing to insufficient NMB reversal following administration of the reversal agent (experimental drug). In case of insufficient reversal, additional administration of sugammadex 2 mg/kg was performed. Another patient was excluded, as his surgery was changed intraoperatively from laparoscopic cholecystectomy to open surgery. As a result, 102 patients participated in the study (49 in Group S and 53 in Group P). The baseline characteristics of the patients were homogeneous (Table 1). Although the female participants in Group S outnumbered their male counterparts, the difference was not statistically significant. The two groups did not exhibit any significant differences in operation times, anesthesia times, amount of remifentanil administered intraoperatively, or amount of fentanyl administered in the PACU (Table 1).
As a primary outcome, the time that elapsed between injection of the NMB reversal agent and the first gas-out was compared between the groups. Group S took 15.03 (6.36–20.25) h, and Group P took 20.85(16.34–25.86) h (P = 0.001) (Table 2). The sugammadex group took less time, and the difference was statistically significant.
Since cholecystectomy patients are usually discharged between POD 2 and POD 5, some of the study participants left the hospital before their first defecation time was recorded. We were able to check the defecation records of 28 of 49 patients in Group S and 28 of 53 patients in Group P. We found no significant difference between the groups (P = 0.694) (Table 2).
Group P took 47.26 (38.72–68.54) h, and Group S took 38 (25.07–64.74) h to achieve their first defecation (P = 0.087). Despite the shorter duration associated with Group S, the difference was not statistically significant (Table 2). Our analysis of stool types showed no significant differences between the groups (Table 2). Differences in the incidence of adverse effects, namely nausea and vomiting, were also not significant. Dry mouth, on the contrary, was experienced by five patients in Group S, whereas 17 patients in Group P reported experiencing the same. This difference was found to be significant (Table 3).
Discussion
The findings of this study showed that sugammadex, used as a reversal agent in postoperative patients who underwent surgery under GA, resulted in a quicker recovery of patients’ GI motility compared to a pyridostigmine-glycopyrrolate mixture.
This result differs from previous studies. Sen et al. were expected to improve bowel movements in patients undergoing thyroidectomy due to neostigmine. There was no difference in gas-out times between the sugammadex and neostigmine groups because of increased gastric emptying due to the affinity of steroid hormones for sugammadex. However, our study was based on the hypothesis that glycopyrrolate would predominate in terms of the effect on bowel movement when glycopyrrolate and pyridostigmine are injected simultaneously. The opposite action of pyridostigmine and glycopyrrolate may not be completely offset due to the difference in onset time and duration. Therefore, the use of sugammadex, which does not affect bowel movements, may have a positive effect on postoperative bowel movements compared to pyridostigmin/glycopyrrolate.
This finding is based on patients’ reports of their first postoperative passage of flatus. The finding can also be interpreted to represent a more natural postoperative recovery of GI motility, since the use of sugammadex does not affect patients’ bowel movements or peristalsis. However, we need to consider the conflicting effects on intestinal motility of the pyridostigmine-glycopyrrolate combination. In this regard, we may assume that the anticholinergic effects of glycopyrrolate on bowel movements can overcome the cholinergic side effects of pyridostigmine. One study has reported that neostigmine can promote GI motility in cases of postoperative ileus .
Another study found that AChEIs such as neostigmine and pyridostigmine are effective for acute colonic pseudo-obstruction and not ileus induced by mechanical bowel obstruction . Both these studies indicate that AChEIs can increase bowel motility. Additionally, we found a previous study reporting that the concurrent use of neostigmine and atropine increased GI motility; however, the study design did not compare the drug mixture with any other agents. Furthermore, that study only investigated the impact on bowel movements based on the timing of atropine administration before neostigmine injection .
We acknowledge the slight differences between the published studies that we examined for our study. The duration of action associated with glycopyrrolate is 2–4 h, and that associated with pyridostigmine is longer than 2 h, which may lead to anticholinergic effects on bowel movements . A number of previous studies have confirmed that the use of sugammadex for the reversal of NMB agents can lead to fewer incidents of respiratory complications, residual neuromuscular block, and PONV compared with the use of AChEIs [5,6]. The relevant literature also lists the advantages of the drug in terms of recovery of the cardiovascular system, urinary system, and other systems [23,24].
Based on the above findings, we may expect that sugammadex will have positive effects on the recovery of GI motility when used as an NMB reversal agent for patients who underwent surgery under GA and can help to decrease postoperative ileus. For prevention of postoperative ileus, a variety of approaches have been explored: gum chewing to induce a stimulatory effect; early mobilization that can reduce insulin resistance and have stimulatory effects; laparoscopic surgery that minimizes tissue trauma and bowel handling to reduce inflammatory reactions; use of non-steroidal anti-inflammatory drugs to reduce inflammatory reactions and opioid sparing; and early enteral nutrition and other similar regimens. Still, further benefits may be obtained with the use of comprehensive, multi-faceted approaches . The use of sugammadex can be one such approach. Sugammadex is believed to enable faster postoperative nutrition and decrease GI complications such as constipation and postoperative ileus. These effects lead to reduced length of stay (LOS), which in turn contributes to ERAS® .
Notably, we did not find any significant inter-group differences in terms of time elapsed until the first defecation reported by the patients. This is considered to be the limitation of our study due to the small number of samples. We attribute this lack of significant differences to the data loss caused by a relatively shorter LOS associated with laparoscopic cholecystectomy; a large number of patients left the hospital without reporting the first postoperative defecation within the LOS. This lost data resulted in a smaller sample size (n = 56) (Table 2). With a longer LOS and/or post-discharge phone interviews, we might have secured sufficient data on defecation times, which may have yielded statistically significant results. Inclusion of a larger sample of patients who remain committed to study participation until the time of their first postoperative defecation might have led to a significant difference in the types of stools.
Neostigmine (AChEIs) is known to increase the incidence of nausea and vomiting. However, its concurrent use with atropine or glycopyrrolate does not increase this incidence . Controversial findings have been reported indicating that AChEIs can increase the risks of nausea and vomiting . As indicated in the above research findings, differences in the incidence of nausea and vomiting were not statistically significant. Considering that the primary outcome of the study was not PONV, other risk factors (e.g., sex, smoking, and history of PONV) that could have been induced were not controlled by the study design. Hence, we see some difficulty in acknowledging the accuracy of the findings. Glycopyrrolate is associated with potent inhibition of salivary gland and respiratory secretions . A significant difference in terms of dry mouth incidence was found in the pyridostigmine group.
The type of surgery targeted may also be a limitation of this study. Laparoscopic cholecystectomy, the focus of our study, involves less handling of the bowel and has fewer effects on bowel movements. Future studies should investigate other types of procedures, such as gastrointestinal surgery and colorectal surgery, which directly influence bowel movements due to the bowel handling and anastomosis involved. Using these surgical procedures, clearer outcomes may emerge in the recovery of GI motility in patients who undergo surgery under GA and are administered with the two reversal agents [29,30].
Furthermore, measuring gastrointestinal transit time by using a scintigraphic method with radioisotopes attached to drugs will likely enable a more accurate comparison of sugammadex against conventional reversal agents.
In conclusion, for patients undergoing laparoscopic cholecystectomy surgery under GA, the use of sugammadex as an NMB reversal agent resulted in an earlier first postoperative passage of flatus compared with the use of a mixture of pyridostigmine and glycopyrrolate. These findings suggest that the use of sugammadex has positive effects on the recovery of postoperative GI motility.
Notes
Conflicts of Interest
No potential conflict of interest relevant to this article was reported. | https://ekja.org/journal/view.php?number=8590&viewtype=pubreader |
Short for BC by abbreviationfinder, the Canadian province of British Columbia is located on the coast of the Pacific Ocean. To the northwest lies Alaska, north of the Yukon, east of Alberta and to the south the US states of Idaho, Montana and Washington border on British Columbia.
The capital of the province is Victoria. British Columbia has an area of 944,735 square kilometers and a population of 4.4 million. That is 4.8 inhabitants per square kilometer. The largest city in the English-speaking province is Vancouver. Vancouver is the economic center of the region.
In the east of the province there are four mountain ranges of the Rocky Mountains. At 4663 meters, Mount Fair weather is the highest mountain in British Columbia and is located in the northwest. The Fraser River is the longest river in British Columbia at 1,375 kilometers. It flows into the Pacific. The famous Vancouver Island is located off the coast.
Climate in British Columbia
There is an arctic climate in the north of the province. The annual average temperature in Fort Nelson is -1.0 ° C. In the months of December and January, mean temperatures can fall below -20 degrees Celsius. At 145 centimeters, northern British Columbia fell in 1999, the largest ever measured amount of snow per day. In the months of June, July and August, temperatures rise to up to 23 ° C during the day.
In the south and on Vancouver Island there is a warm, temperate climate due to the Kuroshio Current in the Pacific Ocean. In this region in British Columbia (Canada) daytime temperatures of around 20 ° C can be reached in summer. The amount of precipitation is pleasantly low in the summer months. In winter, temperatures around freezing point predominate in the south, especially at night, with regular snowfalls.
When to go to British Columbia
In the flat north-east of the province, the arctic climate is much less noticeable in the summer months of June, July and August. Temperatures climb up to 23 ° C during the day and drop to around 9 degrees at night. The best time to visit the Fort Nelson region is June, July and August.
The capital Victoria is located at the southernmost end of British Columbia and, like the popular travel destination Vancouver, has a warm, temperate climate. The daytime temperatures are usually positive all year round, but frost can still occur between November and March.
Summer is considered the best time to travel to British Columbia, the season starts around mid- May and runs until the beginning October. It gets particularly busy in July and August, and the Indian summer (usually at the end of September) also attracts numerous visitors. Winter is all about winter sports, especially the famous Whistler Blackcomb boot ski area between December and early March.
Optimal travel time for the regions
In the following overview you can see the best travel time depending on the region (British Columbia).
|Place||Best travel time|
|Vancouver||May, June, July, August and September|
|Victoria||May, June, July, August and September|
|Fort Nelson||June, July and August|
Temperatures, precipitation, sunshine in Vancouver (British Columbia)
|Jan||Feb||March||Apr||May||Jun||Jul||Aug||Sep||Oct||Nov||Dec|
|Daytime temperature||8 ° C||9 ° C||11 ° C||15 ° C||17 ° C||21 ° C||23 ° C||23 ° C||20 ° C||16 ° C||10 ° C||8 ° C|
|Night temperature||1 ° C||1 ° C||3 ° C||6 ° C||9 ° C||12 ° C||14 ° C||14 ° C||11 ° C||7 ° C||3 ° C||1 ° C|
|Water temperature||8 ° C||8 ° C||8 ° C||10 ° C||11 ° C||13 ° C||14 ° C||14 ° C||13 ° C||11 ° C||9 ° C||8 ° C|
|Precipitation in mm||186||94||118||92||60||59||39||40||48||127||183||178|
|Rainy days||22||16||19||16||12||10||8||7||9||16||21st||20|
|Hours of sunshine||3||4||4||5||7||8||10||9||7||6||4||4|
|Sunrise||8:00||7:20||7:30||6:20||5:30||5:05||5:20||6:00||6:50||7:35||7:20||8:00|
|Sunset||16:40||17:30||19:15||20:05||20:50||21:20||21:15||20:30||19:25||18:25||16:30||16:15|
Temperatures
The annual average temperature in Vancouver is 11 ° C. For comparison: Munich reaches an average of 8.6 ° C, in Berlin it is 9.6 ° C. The warmest month is July (18.5 ° C), coldest month of January with mean values of 4.5 ° C. The maximum water temperature in Vancouver is 14 ° C. That is hardly suitable for swimming.
Precipitation
The annual precipitation is 1224 mm on 176 days with precipitation. For comparison: In Munich, 967 mm, in Berlin 570 mm, precipitation is measured annually. The months of January, November and December are considered the rainy season, as more than 175 mm of precipitation can be expected per month. Overall, about 547 mm of precipitation can be expected in the rainy season. In the months of January, February, March, November and December is possible snowfall.
Sunshine
As our climate table shows, July is the sunniest month with an average of 10 hours of sunshine per day. On average over the year, the sun shines 5.9 hours per day. For comparison: In Munich and Berlin, the sun shines an average of 4.7 hours per day throughout the year.
Cities nearby
The following major towns are nearby and have similar climates: Surrey, Burnaby, Ladner, British Columbia, Richmond, Coquitlam, and Delta.
Highlights and attractions
British Columbia has a lot to offer in terms of activities in particular. The wide and rocky landscapes are ideal for hiking, mountaineering and climbing. You can go canoeing, boat trips or fishing in various waters. There are also opportunities to raft in white water. Many sports facilities can be visited from the 2010 Winter Olympics in Whistler and Vancouver and some are even open to the public.
British Columbia weather in March, April and May
Average daily temperatures between -1 ° C and 17 ° C can be expected over the next three months. May is the mildest in Vancouver, while Fort Nelson is noticeably colder in March. Temperatures in Vancouver are between 11 and 17 ° C, in Victoria between 10 and 16 ° C and in Fort Nelson between -1 and 17 ° C.
Do you want to go on a beach holiday? The water temperatures are in March, April and May 8-11 ° C. So the weather is not suitable for swimming.
In March it rains depending on the region of 6 (Fort Nelson) to 19 days (Vancouver), in April to 5 (Fort Nelson) to 16 days (Vancouver) and in May to 7 (Fort Nelson) to 12 days (Vancouver).
In the period from March to May the sun shines on average between 4 and 9 hours a day. The sunniest weather is in Victoria in May, but with less sun you will have to get by in Vancouver in March. | https://www.bridgat.com/british-columbia-canada-weather-and-climate/ |
Research Proposal Applications are Due August 1st, 2021
We are now calling for research proposal applications for 2022-2023!
Secure funding for your floriculture research project by applying before the deadline of August 1st. We moved the deadline back from June to August to be mindful of busy academic calendars during May and June. This updated deadline will give our researchers more time to work on their proposals.
The primary research priorities are listed below. They are focused on all floricultural crops – fresh-cut flowers, fresh-cut greens, flowering potted plants, foliage plants, and bedding plants.
- Botrytis Control and Management
- Thrips Control and Management
- Biocontrol of Pests
- Postharvest Technology
- Production Technology
- Advanced Breeding Technology, including CRISPR
- Long-Term Storage and Shipping Conditions for Cut Flowers, Bedding and Potted Plants
- Automation and Technology Leading to Labor Savings
- Sustainable Production and Handling Practices
- Reduce the Impact of Climate Change on Production, Handling, and Product Quality
As a reminder, the application process allows for a single application form to be used. This updated application streamlines the researcher’s request for funding.
Projects can last from one to three years and any reasonable but justifiable budget will be considered.
For the current 2021-2022 cycle, the Endowment is funding more than $600,000 in research for 7 new projects and 6 continuing projects.
AFE’s currently-funded research is available online, as well as final reports from previously funded projects. Have suggestions or need information on specific problems? Contact AFE’s Research Coordinator, Dr. Terril A. Nell ([email protected]).
Apply here and submit all supporting documents by the August 1 deadline. | https://endowment.org/new-deadline-afe-research-proposal-applications-2022-2023/ |
Tell us about yourself! What language are you learning and why did you choose to learn it? What are your goals?
I decided to start learning Hebrew a little over a year ago. I've always been interested and curious about languages. Growing up in a Franco-German bilingual environment, I later on added English to my list of interests. Learning Hebrew is a challenge because it's neither a Germanic nor a romance language. I wanted to challenge myself by learning something to which I had never had exposure.
My goal is simple: being able to have a basic conversation with native speakers, while also being to get around when traveling to Israel. It's important to me to try to speak a little bit of the language when traveling to a foreign country, versus expecting them to speak English.
What’s your typical day like and how does Fluent City fit in? Do you do anything else to incorporate language learning into your daily life?
What is a typical day anymore? 2020 has definitely changed our definition of what a typical day is. I'm fortunate and grateful to still have a job. So of course, my primary focus is my job.
Fluent City is a pleasure at the end of the day or during the weekend. Thanks to my teacher’s great flexibility and understanding, I'm able to have a lesson a week, which is always something I look forward to. I wish I could say that I do many extra language-learning activities during the day, but unfortunately, I don't have a lot of time to do so. I do what I assume many other students do: homework, tv shows, and music.
How do you think learning a language will change your connection with others?
Learning a new language is also learning a new culture. It's important to learn both, as it's a tool to understand others. Understanding others is what helps us grow as individuals and communicate respectfully.
I've always been curious about different cultures and languages. Learning a new language is also learning a different version of yourself. When you speak a different language, you communicate differently and show a different self to others. All that because of the words and syntax you use.
What impact has learning a language had on your life in the last few months?
Learning a language had an impact on my routine. Instead of watching tv and doing nothing during the quarantine at night, my brain stayed active and I felt challenged on a weekly basis. I never experienced boredom because I always had something to do during my down time. It also made me more curious about culture and history.
What's been your favorite socially distant activity to do with friends and family over the last few months?
Probably similar to other people - Zoom meetings with my friends and family. When things started reopening, Central Park picnics with social distancing. One thing I experienced during the last few months is how to reconnect/communicate with important people in my life. I realized that we used to take things for granted, and forgot how to actually talk to people. | https://blog.fluentcity.com/how-hebrew-helped-nathan-avoid-quarantine-boredom/ |
Minister of Communication, Barrister Adebayo Shittu, weekend disclosed that Nigeria’s revolution in the ICT sector accounts for over $32 billion in foreign direct investment over the last 15 years.
The Minister noted that investments in infrastructure has produced an ICT backbone that powers various critical sectors of the economy such as Banking, E-commerce, Insurance, and Oil and Gas.
Speaking at the one-day conference themed ‘Africa-China Cooperation in ICT and Digital Economy’ in Abuja, organized by the Nigerian Institute of International Affairs in collaboration with the Embassy of China in Nigeria and Huawei Technologies Company (Nigeria) Limited, Shittu advocated the need for all stakeholders to strengthen the technology and innovation ecosystem.
He affirmed that the fortification and supporting the development of innovation hubs in partnership with the private sector will be beneficial to all.
“This conference provides the opportunity to jump-start the critical game-changing steps needed to make Nigeria’s objectives a reality in digital economy, he said
“Beyond this conference, we must work to strengthen relationships and knowledge management platforms towards building the better and more digital future that we seek.”
Special Advisor on ICT to the Vice President, Lanre Osibona, at the conference said that there is an urgent need to overcome challenges facing the nation to leveraging ICT to fuel the fourth industrial revolution that brings about digital economy.
According to him: “Africa must develop its skills. We must know how to scale workforce and move away from business base outsourcing to knowledge base outsourcing. Data is the future and new hope. For us as a country we must invest heavily in capturing data.
The Ambassador of China to Nigeria, Dr. Zhon Pingjian, on his part asserted that “China will continue to share its development opportunities with African countries and welcome them on board the train of China’s development.”
For Tank Li, Managing Director, Huawei Technologies Nigeria, a robust ICT infrastructure is the bedrock for digital transformation in Nigeria, and to unleash Digital Economy potentials in the country, issues of availability and affordability need to be addressed.
“In order to foster digital transformation of the economy, policies and programs to increase ICT infrastructure and ensure wide-spread coverage both in urban and rural areas should be prioritized to make voice and data services available. Li said
“At the same time, strategic measures of infrastructure sharing, investment-friendly regulatory framework and preferential taxation policies are needed to reduce sites acquisition and broadband deployment costs in order to bring down the cost for users to really encourage application of ICT across the industries and the whole society.”
According to a study by PWC, constrained digital economies can potentially realize a 0.5 percent increase in GDP per capita for every 10 percent increase in digitization.
The study also highlights that digitization also has a significant impact on job creation in the overall economy: an increase of 10 percent in digitization reduces a nation’s unemployment rate by 0.84 percent.
The conference featured discussions on policy issues with a view to propose a domestic approach to not only enhancing the strategic priority of ICT and penetration of fixed broadband, but also improving connectivity through cost-effective network deployment, and easing the development of local content such as e-commerce in accelerating economic growth.
A request by the stakeholders at the conference was made, for a framework and policies from the government that will foster broadband infrastructure development in Nigeria to help stimulate GDP growth.
A necessity to continue to collaborate with private sector and foreign investors that help to drive innovation for scaling ICT in Nigeria was also noted.
Digitization, widespread adoption of digital technologies and applications by Corporates, Government and Consumers, is beneficial for an emerging economy like Nigeria. | |
The land mass of the planet is in a constant state of flux and is divided into four empires, each one claimed by one of the Chaos powers who wage eternal wars to encroach into other territories.
With the victory of the Crusaders of Darkness, the planet of Vordrast, now a daemon world, has been pitted into perpetual war amongst the four major Chaos Gods. Armies of daemons and their living allies fight huge and bloody battles to determine which of the Chaos Powers will dominate Vordrast, adding the world to their realms within the Warp. Belakor's decision has laid waste to the planet as entire hive cities have become little more than gigantic arenas where the opposing forces are pitched against each other.
Unfortunately several abandoned forces unable to escape the planet struggle daily to survive except for the Necrons who have erected strange pylons as a defence against the psychic abilities of the warp. With no possible refuge Imperial Marines, Eldar, and Guard units prowl through the ruins eager to uncover abandoned warp capable vessels that have escaped the notice of the dark gods during the chaos struggles. | https://www.leicesterallscars.org/return-to-vordrast.html |
Foreign ministers of four-member bloc are meeting in Cairo to weigh Qatars response
Companies including banks and financial institutions, and some trade partners of the UAE, Saudi Arabia and Bahrain are anxiously awaiting the latest twists in the ongoing political rift between Qatar and the bloc of four Arab nations to unfold, which could have serious consequences for their business relations with Doha.
Qatar is engaged in a diplomatic spat with Arab states that have accused it of funding terrorism and destabilising the region. Gulf monarchies including Saudi Arabia, the UAE, Bahrain and their allies Egypt, Libya, and Yemen moved to cut diplomatic, as well as trade and transport links, with Qatar on 5 June. The economic measures imposed include the closing of the land border with Saudi Arabia, a blockade of sea and air access, and the expulsion of Qatari officials, residents and visitors.
The bloc of four countries has jointly named 12 entities and 59 individuals who are funding or supporting terrorists or terrorist organisations, and has presented a list of demands to Qatar, which includes shutting down Doha-based Aljazeera TV, reducing ties with Iran and shutting down a Turkish military base in Qatar.
Qatar has denied the allegations and rejected the demands from the outset, saying it will not negotiate while the economic blockade remains in place. Its foreign minister, Sheikh Mohammed bin Abdulrahman al-Thani, who later said Doha was ready to engage in dialogue under the right conditions, this week delivered a letter to officials in Kuwait, which is acting as a mediator in the political crisis.
Qatar was given an initial deadline of 3 July to respond, which was later extended by 48 hours.
It is not known what Dohas response has been to demands made by the Saudi-led alliance, but there are indications that a rejection from Qatar would entail further economic sanctions.
The foreign ministers of Saudi Arabia, the UAE, Bahrain and Egypt are meeting on 5 June in Cairo to discuss Qatars response and consider further actions against it.
The UAEs ambassador to Moscow, Omar Ghobash, in the last week of June said the Gulf states already were considering further economic pressure such as reducing commercial links with countries that continue to trade with Qatar.
One possibility would be to impose conditions on our own trading partners and say if you want to work with us then you have got to make a commercial choice, Londons Guardian newspaper quoted Ghobash as saying in an interview.
Public exchanges between the different parties in recent weeks suggest a quick resolution is unlikely and that the stalemate may continue for some time, according to a 4 July statement from US-based Moodys Investors Service. There is also the possibility of a restriction placed on banks based in the four-member bloc in terms of doing business in Qatar and with Qatari entities. The banks have already been asked to seize the assets and accounts of individuals and entities with terror-related links, and regulators in the UAE and Bahrain have told lenders to exercise caution when dealing with six Qatari banks.
Sources have indicated to MEED that there could be further restrictions on banks in the coming weeks and months.
Weaker economic activity could also lead to deteriorating asset quality in the banking system and together with an escalation involving sanctions against the financial sector could necessitate a step-up in government liquidity support, said Moodys.
The banking system, which experienced intense volatility following the announcement of the blockade against Qatar, is now seeing a recovery and normalisation of activities. But there could be more instability in store for lenders in the region if the foreign ministers meeting in Cairo decide to up the ante and impose banking sanctions on Qatar as well.
You might also like...
A MEED Subscription...
Subscribe or upgrade your current MEED.com package to support your strategic planning with the MENA region’s best source of business information. Proceed to our online shop below to find out more about the features in each package. | https://www.meed.com/companies-await-latest-twists-in-gcc-rift |
A broken wrist is common following a fall on an outstretched hand. A Colles fracture is a fracture of the Radius bone of the forearm, just above the wrist (a Scaphoid fracture is the other common type of wrist fracture). Symptoms include a great deal of wrist pain, a “dinner fork” deformity, wrist swelling and an inability to use the wrist and hand. The term Colles fracture originated with the Dublin doctor Abraham Colles, who first described this common type of wrist fracture in 1814.
Falls that cause wrist fractures are common during skiing and snowboarding in particular where falls at high speed are common. The natural response to a fall is to stretch out a hand to break the fall, and falls tend to occur more often in beginners. For this reason a broken wrist is a relatively common feature, with around 100,000 wrist fractures worldwide among snowboarders each year.
Broken Wrist Signs & Symptoms
As you would imagine with a fracture, there is a great deal of wrist pain. The aforementioned ‘dinner fork’ deformity is usually present, named so as it resembles a dinner fork when viewed on Xray from the side. Together with wrist swelling and bruising, as a result of the fracture, there is an inability to use the wrist and hand. In most Colles fractures the patient will hold the affected wrist towards their body in an effort to protect it. Diagnosis is confirmed by an x-ray.
Broken Wrist Treatment
What you can do
|Consult a sports injury expert|
|Apply cold therapy to relieve pain|
|Wear a removable wrist support for protection|
|Regain dexterity with therapeutic putty|
|Relieve wrist stiffness with hand therapy balls|
|Improve hand & grip strength with resistance exercises|
|Use a bone healing system to speed up broken bone healing|
If a fracture of the wrist is suspected, the patient should be taken to an accident and emergency department without delay. Ice Packs may be helpful to relieve pain, reduce swelling and limit bruising. .
If a Colles fracture is confirmed on x-ray, the initial treatment will be supervised by the doctor in the emergency department. If the two fragments of broken bone are shown to be close together, and well aligned, the treatment is simply to immobilise the wrist in a cast for 6 weeks. For the first 72 hours after the injury the wrist should be elevated to reduce swelling.
However, if the x-ray shows that the two fragments of bone are displaced away from each other, or not well aligned, then the emergency department doctor will have to manipulate the fragments back into position. This procedure is done under anaesthetic and once the bone has been repositioned, the forearm and wrist will be immobilised in a plaster for 6 weeks. The wrist should be x-rayed again after 2 weeks to make sure that the bone is still well aligned. Depending on the preference of the treating doctor it may be possible to use a Removable Cast from two or three weeks so long as the x-ray shows that the bone is healing well.
Removable Wrist Supports provide the same degree of protection as a conventional plaster, but are much lighter and therefore more comfortable to wear for day to day activities. The fact that they can be removed allows washing of the wrist region, making it much more hygienic than a conventional plaster which allows the user to keep the skin in good condition.
Rehabilitation begins immediately by maintaining the range of movement in the shoulder, elbow, fingers and thumb, on the side of the affected wrist. This prevents secondary stiffness in these areas and helps to resolve swelling in the wrist. Assuming that there are no complications with healing, the plaster can usually be removed after 6 weeks, if the doctor is satisfied that the bone has united and healed itself. At this stage more active rehabilitation can be undertaken.
Exercises in warm water are helpful to improve the hydration of the skin if it was encased in a plaster of paris. These also encourage the patient to gently begin moving the wrist in all directions, relieving stiffness. Exercises using Therapeutic Putty, Hand Therapy Balls and Finger & Grip Strengtheners can add strength to the intrinsic muscles around the hand and wrist and resolve wrist swelling. In the period following the removal of the plaster it may be helpful to wear a Wrist Support when not doing the exercises and when reintroducing activities that involve wrist movement.
Broken Wrist Prevention
Snowboarders should wear wrist guards as they significantly reduce the incidence of wrist injuries during falls. These are available from all good ski shops. | https://physioproductskenya.com/injury-help/broken-wrist-in-depth-aka-colles-fracture/ |
By Re-Nuble Team
Lena is one out of two of our summer interns that our team was blessed with this year. Hailing from The City College of New York, Lena's energy for all things sustainable and transparent really resonated with our Founder when the two met during CUNY's Internship Fair. Lena's strong facility to communicate a story that resonated with her personal values and advocacy spoke highly of her passion for wanting to tell a story that was consciously right and not trend driven. For this, reason we've enjoyed Lena's contribution to the team to date!
1. What inspired you to want to work with Re-Nuble?
I was inspired to work for Re-Nuble because I have always felt strongly about sustainability within public policy and appreciated Re-Nuble’s ability to maintain sustainable practices within a business. As a business-oriented college student, I knew that I would be entering an industry that did not always pride themselves on sustainability. Therefore, I was ecstatic to receive the opportunity to work for a company like Re-Nuble, one that emphasizes the importance of sustainable practices within their business model.
2. Do you have a green thumb and how do you practice it?
Yes! I have always appreciated the beauty that our surroundings have to offer us. I continuously find myself picking up trash wherever I go and am saddened by the thought of people being so careless. I still find it appalling that one can so easily treat our environment with such little care or appreciation; if they won’t pick it up then I will! However, being a green thumb is not necessarily about picking up garbage off the sidewalk but rather a continuous state of mind. I love to help my grandmother garden in upstate New York and grew up having fresh vegetables from our garden. I believe the most important aspect of being a green thumb is not only maintaining your personal eco-friendly practices, but also educating others on how to do the same.
3. What does a Sustainable NYC mean to you and how do you envision Re-Nuble fitting into that vision?
A sustainable NYC to me means a city in which we do not litter on the sidewalks or within parks, have reduced landfill and are eating all organic through sustainable practices. I envision Re-Nuble giving wholesale food distributors the opportunity to decrease the amount of landfill buildup from their business, while promoting the importance of sustainability. This could lead to the use of Re-Nuble’s fertilizer in all urban farms and distributing all organic produce within major grocery stores. I believe Re-Nuble is a company that can bring NYC into a new stage of existence, through educating and inspiring others to think about the environment with every decision they make. A sustainable NYC could become a leading example for other cities throughout the country, to instill sustainable practices as much as possible.
4. How do you practice aspects of a circular economy/sustainability at home?
At home, I collect the clothing that does not fit me anymore or that I am not wearing too often and take them to a thrift store. In return for exchanging my clothing, I receive store credit and use it to purchase more thrifted clothing when I need. I find this process to be something that many might not consider first hand to be a sustainable practice, but it is! By recycling my clothing, I am able to pass along clothing to another individual without increasing my carbon footprint or creating the need to produce new clothing.
5. What would you like to see Re-Nuble do differently for cities that other companies or brands have failed to do so or have not invested the efforts in creating?
I would like to see Re-Nuble bring their vegetable-waste based fertilizers to all urban gardens within major cities. This will provide farmers with the opportunity to maintain sustainable practices and decrease the amount of landfill from major cities. In tandem with this, I see Re-Nuble providing major cities with the ability to grow and eat all organic. This could change the way society looks at the quality and taste of our food; in doing so promoting organic and healthy eating while remaining sustainable. I would like to see Re-Nuble be the forefront of the industry because they are genuine in their mission, just like their products. | https://www.re-nuble.com/blogs/re-nuble/meet-lena-quinn-our-business-associate-summer-intern |
Gaithersburg MS is located at 2 Teachers' Way, Gaithersburg, MD 20877. The principal is Mrs. Ann Dolan Rindner.
RatingPlease rate your experience at this school with respect to the following aspects on a scale from 0 to 10.
- Inclusiveness: Does the school have a culture of inclusion and inclusive practices in place (e.g., co-teaching)? Are these practices effectively and systematically implemented?
- Staffing and resources: Does the school have staff and resources to accommodate a wide range of special needs?
- Cooperation: Does the school staff exhibit a cooperative approach by accepting/requesting the parents' input, listening to concerns, and attempting to resolve disagreements?
- Communication: Does the school implement progress monitoring plans? Does the staff regularly communicate with parents with respect to their child's progress and challenges? Is the staff responsive when contacted?
- Academic expectations: Does the staff systematically set high expectations for students with special needs? Does the staff systematically raise expectations when IEP goals are met ahead of time? | http://www.r2minnovations.com/mcps-reviews/schools/public-schools/middle-schools/gaithersburg-ms/ |
Q:
AES encryption for hex key, IV and data in Java
I have to encrypt the hex string with key and IV also as Hex data. Please find the code i have used below. The problem is the string which i need to encrypt is in plaintext. I need to convert the plaintext into Hex and i do not know how to do that. Can any one help me with it ??
I have been confirming using the webtool,
http://aes.online-domain-tools.com/
static byte[] IV = { 0x00, 0x01, 0x02, 0x03, 0x04, 0x05, 0x06, 0x07, 0x08, 0x09, 0x0a, 0x0b, 0x0c, 0x0d, 0x0e, 0x0f };
static String plaintext = "00010101010102020202020303030303";
static byte[] encryptionKey = {0x41, 0x41, 0x41, 0x41, 0x41, 0x41, 0x41, 0x41, 0x41,
0x41, 0x41, 0x41, 0x41, 0x41, 0x41, 0x41 };
public static void main(String[] args) {
try {
System.out.println("==Java==");
System.out.println("plain: " + plaintext);
byte[] cipher = encrypt(plaintext, encryptionKey);
System.out.print("cipher: ");
for (int i = 0; i < cipher.length; i++) {
// System.out.print(new Integer(cipher[i]) + " ");
byte b = cipher[i];
System.out.print(String.format("%02x", b & 0xFF) + " ");
}
String decrypted = decrypt(cipher, encryptionKey);
System.out.println("decrypt: " + decrypted);
} catch (Exception e) {
e.printStackTrace();
}
}
public static byte[] encrypt(String plainText, byte[] encryptionKey)
throws Exception {
Cipher cipher = Cipher.getInstance("AES/CBC/NoPadding");
SecretKeySpec key = new SecretKeySpec(encryptionKey, "AES");
cipher.init(Cipher.ENCRYPT_MODE, key, new IvParameterSpec(IV));
return cipher.doFinal(plainText.getBytes("UTF-8"));
}
public static String decrypt(byte[] cipherText, byte[] encryptionKey)
throws Exception {
Cipher cipher = Cipher.getInstance("AES/CBC/NoPadding");
SecretKeySpec key = new SecretKeySpec(encryptionKey, "AES");
cipher.init(Cipher.DECRYPT_MODE, key, new IvParameterSpec(IV));
return new String(cipher.doFinal(cipherText), "UTF-8");
}
for the above data i should get,
b6 06 da b9 cd dc 2d 89 4b 49 0a ab 4e e7 dc 58
but instead am getting,
e3 62 34 3f ad 8b 89 37 57 81 91 31 ee 79 49 52 26 bf 40 cb d0 ce 36 bd 8a 04 6b af 34 d9 f3 d7
Thanks in Advance.
A:
Probably the best thing to learn from this is that modern modes of operation such as CBC (as well as the underlying block cipher) operate on binary data, usually provided as a set of bytes.
In that sense it is probably best to stick to binary data as much as possible. Don't use more conversions than strictly necessary and only use hexadecimal encoding if it is meant for human consumption. Base 64 encoding can be used if you need to transfer binary using a text based representation.
Encoding/decoding should preferably be performed using a library such as Apache Commons Codec, Guava, Bouncy Castle etc., Java 8 still doesn't hex encode/decode byte arrays (sigh).
I've created a small code sample that outputs hex formatted as if it was a Java byte array literal. I'll leave it to you to create different versions.
static byte[] IV = { 0x00, 0x01, 0x02, 0x03, 0x04, 0x05, 0x06, 0x07, 0x08, 0x09, 0x0a, 0x0b, 0x0c, 0x0d, 0x0e, 0x0f };
static byte[] plaintext = { 0x00, 0x01, 0x01, 0x01, 0x01, 0x01, 0x02, 0x02, 0x02, 0x02, 0x02, 0x03, 0x03, 0x03, 0x03, 0x03 };
System.out.println("plain: " + toJavaHex(plaintext));
byte[] cipher = encrypt(plaintext, encryptionKey);
System.out.println("cipher: " + toJavaHex(cipher));
String decrypted = toJavaHex(decrypt(cipher, encryptionKey));
System.out.println("decrypt: " + decrypted);
} catch (Exception e) {
e.printStackTrace();
}
}
public static String toJavaHex(byte[] data) {
StringBuilder sb = new StringBuilder(13 * data.length);
for (int i = 0; i < data.length; i++) {
if (i != 0) {
sb.append(", ");
}
sb.append(String.format("(byte) 0x%02x", data[i]));
}
return sb.toString();
}
public static byte[] encrypt(byte[] plainText, byte[] encryptionKey)
return cipher.doFinal(plainText);
}
public static byte[] decrypt(byte[] cipherText, byte[] encryptionKey)
return cipher.doFinal(cipherText);
}
| |
# Otto-Dix-Haus
The Otto-Dix-Haus or Otto Dix House in Gera is a museum located in the birth house of the German painter Otto Dix, at Mohrenplatz 4. The building became an art museum in 1991, with exhibits on two floors. for the celebrations of the 100th birthday of Otto Dix. The museum, with the Orangerie, is part of the Kunstsammlung Gera.
## Building history
The half-timbered building was built in the 18th century. In 1946 the rear building was converted for residential purposes. In the run-up to the 100th birthday of Otto Dix, who was born in the house on December 2, 1891, the building was converted into a museum from 1988 to 1991. For this purpose, an extension was built on the west side as an access and functional wing.
## Museum exhibition
The museum evokes the atmosphere of a simple working-class house of the beginning of the 20th century and shows materials about the life and work of the artist. The museum's permanent exhibition includes 400 of his drawings and oil paintings, and printed graphics from its own holdings. In addition to works by Dix, which have a permanent place, there are also changing exhibitions of other loans. In addition to the works of Otto Dix himself, the museum regularly hosts exhibitions by other artists, including photo exhibitions, for which there is a separate exhibition hall. The rooms where Dix lived contain his work. The extensive collection of works on paper is displayed in alternating special exhibitions. After prior notice it is also possible to visit the study room and the collection of graphics of the house. The collection also includes the 48 postcards he sent during his service at World War I.
The paintings from several phases of the artist work exhibited in the museum include: Selbstbildnis als Raucher (1913), Meine Freundin Elis (1919), Doppelbildnis Otto Dix/Kurt Günther (1920), Der Heilige Christophorus IV (1939), and Bildnis des Malers Hans Theo Richter mit Frau Gisela (1933).
In addition to exhibition activities, the house-museum also runs educational courses for teachers and artists, including youth groups divided by age.
## Renovation
On 3 June 2013, the flooding of the White Elster River reached the ground floor of the museum. The flood damage made it necessary to renovate the building. The Otto-Dix-Haus therefore closed on 4 January 2016, and some exhibits were shown in an interim exhibition in the city museum. On Dix's 125th birthday, on 2 December 2016, the museum reopened with the special exhibition "Otto Dix: Drawing Art with Silver Pen". | https://en.wikipedia.org/wiki/Otto-Dix-Haus |
Chances are that you and I learn differently. This is because we all have different learning modalities. There are three different learning modalities: auditory, visual, and kinesthetic/tactile. You may learn via one modality or you may learn primarily one way with a little bit of another modality. In order to figure out how you learn, there are a variety of learning modality tests online that you can take. If you don’t have the time to take a test, here are some explanations of the different learning modalities.
Do you like hearing information being presented to you? Do you talk or hum to yourself? If you answered yes, then chances are you are an auditory learner. Auditory learners learn the best by hearing things. Videos and audio recordings are great ways for this type of learner to learn. This type of learner will sit where they can hear information, but may not always look like they are paying attention. Reading out loud is another strategy for this type of learner. Have students repeat things back to you, or read directions. Auditory learners will learn from other students reading out loud. When studying, this type of learner will study out loud.
Visual learners need to see information in order to understand it. They like to take lots of notes and make detailed lists. The visual learner will highlight information in bright colors when taking notes in order to make it stand out. They sit in the front of the classroom so they can see the information from the nearest seat. They may also try to picture things in their head to remember something. The visual learner also likes to see pictures, charts, diagrams, and illustrations.
Kinesthetic/tactile learners need to experience things hands-on. They learn well from lab experiments, projects, and field trips. This is the most active type of learning modality. They tend to fidget a lot and need to take frequent breaks. In the classroom, make sure they sit away from any distractions as they can be very easily distracted. This type of learner may try to put something together by experimenting instead of looking at the pictures or reading directions. Kinesthetic/tactile learners rely on what they can directly experience.
Now that you know about the different types of learning modalities, you can try adapting your teaching style to meet the needs of all learners. Most teachers are good at presenting information visually and auditorily, however, sometimes we can forget about the hands-on learner. Teachers need to figure out how to incorporate this learning style into their teaching. As you teach lessons, think of how you can get the students up and moving and doing projects in order for them to learn.
These learning modalities need to be addressed every day as you teach your lessons. As you plan lessons, think how you can adapt this lesson to meet the needs of the visual, auditory, and kinesthetic learners. Though you may not teach to every learning style every day, try to change things up in your classroom in order to meet the needs of all of your learners each week. | https://www.brighthubeducation.com/teaching-methods-tips/79946-learning-modalities-auditory-tactile-and-visual/ |
In-vitro regeneration potential for three high beta-carotene (pro-vitamin A) cassava varieties - UMUCASS 36, UMUCASS 37 and UMUCASS 38, and a control variety TMS 60444 were evaluated and optimized as a preliminary step towards the introgression of more nutritional and agronomic traits. Somatic embryos developed from in vitro plantlets were used for the production of callus tissues from which whole cassava plantlets were regenerated. The frequency of somatic embryogenesis (SE) for UMUCASS 36, 37 and 38 were 43% (10/23), 50% (8/16) and 43% (20/47) respectively while that for the TMS 60444 used as control was 64% (38/59). The regeneration efficiency expressed as the percentage of plant lines recovered from total number of cotyledon lines derive from UMUCASS 36, 37, 38 and TMS 60444 were 40% (2/5), 29% (2/7), 38% (3/8) and 67% (10/15) respectively. The regeneration efficiencies for the UMUCASS varieties were less than the control and thus require further optimization of the regeneration protocol for bulking of regenerated plantlets derived from these genotypes. | https://www.ajol.info/index.php/naj/article/view/162699 |
Q:
Are manifolds defined by level sets uniquely determined by gradients?
Let $f : \mathbb R^n → \mathbb R$ be a continuously differentiable function with non-zero gradient. Then, according to the implicit function theorem a level set defines a $n-1$ dimensional manifold $M$ and the gradient of $f$ is perpendicular to $M$.
Now, assume that we have a second function $g : \mathbb R^n → \mathbb R$ also continuously differentiable with non-zero gradient.
How to prove or refute precisely that if for all points in $M$ (defined by the level set of $f$) the direction of the gradient of $f$ and $g$ is identical (but not the length), then $M$ is also a manifold defined by a level set of $g$.
A:
Suppose that $M=f^{-1}(a), a\in R$, for every $x$ in $M$, we have to show that $g$ is constant on the connected component of $M$ which contains $x$. Let $c:I\rightarrow M$ be a differentiable path such that $c(0)=x$, ${d\over{dt}}g(c(t))=\nabla g_{c(t)}.c'(t)=u(t)\nabla f_{c(t)}c'(t)=0$ since the restriction of f to M is constant and $\nabla f$ is proportional to $\nabla g$. This implies that $g$ is constant on $c$ and henceforth on the connected component of $M$ which contains $x$.
| |
Creating a price index
This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!
Create a price index for Green Bay Packer fans using the following basket of goods with 1997 prices as the base year.
Quantities Prices
In 1997 1997 1998
90 lbs of cheese $2.50/lb $2.0/lb
12 flannel shirts $15/shirt $20/shirt
16 football tickets $25/ticket $30/ticket
https://brainmass.com/economics/finance/creating-a-price-index-515768
Solution Preview
Total value of given basket of goods at 1997 prices=90*2.50+12*15+16*25=$805
Total value of given basket of goods at 1998 ...
Solution Summary
Solution describes the steps to create a price index in the given case. | https://brainmass.com/economics/finance/creating-a-price-index-515768 |
---
abstract: 'We suggest a possible approach to proving the Mézard-Parisi formula for the free energy in the diluted spin glass models, such as diluted $K$-spin or random $K$-sat model at any positive temperature. In the main contribution of the paper, we show that a certain small modification of the Hamiltonian in any of these models forces all finite-RSB asymptotic Gibbs measures in the sense of the overlaps to satisfy the Mézard-Parisi ansatz for the distribution of spins. Unfortunately, what is still missing is a description of the general full-RSB asymptotic Gibbs measures. If one could show that the general case can be approximated by finite-RSB case in the right sense then one could a posteriori remove the small modification of the Hamiltonian to recover the Mézard-Parisi formula for the original model.'
author:
- 'Dmitry Panchenko[^1]\'
title: |
Structure of finite-RSB asymptotic Gibbs measures\
in the diluted spin glass models
---
Key words: spin glasses, diluted models\
Mathematics Subject Classification (2010): 60K35, 60G09, 82B44
Introduction
============
This paper continues to develop the line of ideas in [@Pspins; @HEPS; @AP; @1RSB], which are all motivated by the Mézard-Parisi formula for the free energy in the diluted spin glass models originating in [@Mezard]. This formula is closely related to the original Parisi formula [@Parisi79; @Parisi] for the free energy in the Sherrington-Kirkpatrick model [@SK], but at the same time it is more complicated, because it involves a more complicated functional order parameter that encodes some very special structure of the distribution of all spins (or all multi-overlaps) rather than the distribution of the overlaps.
An important progress was made by Franz and Leone in [@FL], who showed that the Mézard-Parisi formula gives an upper bound on the free energy (see also [@PT]). The technical details of their work are very different but, clearly, inspired by the analogous earlier result of Guerra [@Guerra] for the Sherrington-Kirkpatrick model, which lead the to the first proof of the Parisi formula by Talagrand in [@TPF]. Another proof of the Parisi formula was given later in [@PPF], based on the ultrametricity property for the overlaps proved in [@PUltra] using the Ghirlanda-Guerra identities [@GG] (the general idea that stability properties, such as the Aizenman-Contucci stochastic stability [@AC] or the Ghirlanda-Guerra identities, could imply ultrametricity is due to Arguin and Aizenman, [@AA]). The proof there combined the cavity method in the form of the Aizenman-Sims-Starr representation [@AS2] with the description of the asymptotic structure of the overlap distribution that follows from ultrametricity and the Ghirlanda-Guerra identities [@GG].
The Mézard-Parisi ansatz in the diluted models builds upon the ultrametric Parisi ansatz in the SK model, so it is very convenient that ultrametricity for the overlaps can be obtained just as easily in the diluted models as in the SK model, simply because the Ghirlanda-Guerra identities can be proved in these models in exactly the same way, by using a small perturbation of the Hamiltonian of the mixed $p$-spin type. However, as we mention above, the Mézard-Parisi ansatz describes the structure of the Gibbs measure in these models in much more detail, as we shall see below.
Some progress toward explaining the features of this ansatz beyond ultrametricity was made in [@HEPS; @AP], where the so-called hierarchical exchangeability of the pure states and the corresponding Aldous-Hoover representation were proved. This representation looks very similar to what one expects in the Mézard-Parisi ansatz, but lacks some additional symmetry. One example where this additional symmetry can be proved rigorously was given in [@1RSB] for the $1$-RSB asymptotic Gibbs measures in the diluted $K$-spin model, where it was obtained as a consequence of the cavity equations for spin distributions developed rigorously in [@Pspins]. The main contribution of this paper is to show how this result can be extended to all finite-RSB asymptotic Gibbs measures for all diluted models. Namely, we will show that one can slightly modify the Hamiltonian in such a way that all finite-RSB asymptotic Gibbs measures satisfy the Mézard-Parisi ansatz as a consequence of the Ghirlanda-Guerra identities and the cavity equations.
Main results
============
Before we can state our main results, we will need to introduce necessary notations and definitions, as well as review a number of previous results.
Let $K\geq 1$ be an integer fixed throughout the paper. A random clause with $K$ variables will be a random function $\theta(\sigma_{1},\ldots,\sigma_{K})$ on $\{-1,+1\}^K$ symmetric in its coordinates. The main examples we have in mind are the following.
[**Example 1.**]{} ($K$-spin model) Given an inverse temperature parameter $\beta>0$, the random function $\theta$ is given by $$\theta(\sigma_1,\ldots,\sigma_K)= \beta g \sigma_1\cdots \sigma_K,$$ where $g$ is a random variable, typically, standard Gaussian or Rademacher.
[**Example 2.**]{} ($K$-sat model) Given an inverse temperature parameter $\beta>0$, the random function $\theta$ is given by $$\theta(\sigma_1,\ldots,\sigma_K)=-\beta\prod_{j\leq K} \frac{1+J_j \sigma_j}{2},$$ where $(J_j)_{j\geq 1}$ are i.i.d. Bernoulli random variables with ${\mathbb{P}}(J_j=\pm 1)=1/2.$
We will denote by $\theta_I$ independent copies of the function $\theta$ for various multi-indices $I$. Given a parameter $\lambda>0$, called connectivity parameter, the Hamiltonian of a diluted model is defined by $$H_N^{\mathrm{model}}(\sigma) = \sum_{k\leq \pi(\lambda N)} \theta_k(\sigma_{i_{1,k}},\ldots, \sigma_{i_{K,k}})
\label{Ham2}$$ where $\pi(\lambda N)$ is a Poisson random variable with the mean $\lambda N$, and the coordinate indices $i_{j,k}$ are independent for different pairs $(j,k)$ and are chosen uniformly from $\{1,\ldots, N\}$. The main goal for us would be to compute the limit of the free energy $$F_N^{\mathrm{model}} = \frac{1}{N}{\mathbb{E}}\log \sum_{\sigma\in \Sigma_N} \exp H_N^{\mathrm{model}}(\sigma)$$ as $N$ goes to infinity. The formula for this limit originates in the work of Mézard and Parisi in [@Mezard]. To state how the formula looks like, we need to recall several definitions that will be used throughout the paper.
**Ruelle probability cascades (RPC, [@Ruelle]).** Given $r\geq 1$, consider an infinitary rooted tree of depth $r$ with the vertex set $${{\cal A}}= {\mathbb{N}}^0 \cup {\mathbb{N}}\cup {\mathbb{N}}^2 \cup \ldots \cup {\mathbb{N}}^r,
\label{Atree}$$ where ${\mathbb{N}}^0 = \{*\}$, $*$ is the root of the tree and each vertex $\alpha=(n_1,\ldots,n_p)\in {\mathbb{N}}^{p}$ for $p\leq r-1$ has children $$\alpha n : = (n_1,\ldots,n_p,n) \in {\mathbb{N}}^{p+1}$$ for all $n\in {\mathbb{N}}$. Each vertex $\alpha$ is connected to the root $*$ by the path $$* \to n_1 \to (n_1,n_2) \to\cdots\to (n_1,\ldots,n_p) = \alpha.$$ We will denote the set of vertices in this path by $$p(\alpha) = \bigl\{*\, , n_1, (n_1,n_2),\ldots,(n_1,\ldots,n_p) \bigr\}.
\label{pathtoleaf}$$ We will denote by $|\alpha|$ the distance of $\alpha$ from the root, i.e. $p$ when $\alpha=(n_1,\ldots,n_p)$. We will write $\alpha \succeq \beta$ if $\beta \in p(\alpha)$ and $\alpha\succ \beta$ if, in addition, $\alpha\not =\beta$, in which case we will say that $\alpha$ is a descendant of $\beta$, and $\beta$ is an ancestor of $\alpha$. Notice that $\beta\in p(\alpha)$ if and only if $*\preceq \beta\preceq \alpha.$ The set of leaves ${\mathbb{N}}^r$ of ${{\cal A}}$ will sometimes be denoted by ${{\cal L}}({{\cal A}})$. For any $\alpha, \beta\in {{\cal A}}$, let $$\alpha\wedge\beta
:=
|p(\alpha) \cap p(\beta)| - 1
\label{wedge}$$ be the number of common vertices (not counting the root $*$) in the paths from the root to the vertices $\alpha$ and $\beta$. In other words, $\alpha \wedge \beta$ is the distance of the lowest common ancestor of $\alpha$ and $\beta$ from the root.
Let us consider parameters $$0= \zeta_{-1} <\zeta_0 <\ldots < \zeta_{r-1} <\zeta_r = 1
\label{zetas}$$ that will appear later in the c.d.f. of the overlap in the case when it takes finitely many values (see (\[zetap\]) below), which is the usual functional order parameter in the Parisi ansatz. For each $\alpha\in {{\cal A}}\setminus {\mathbb{N}}^r$, let $\Pi_\alpha$ be a Poisson process on $(0,\infty)$ with the mean measure $$\zeta_{p}x^{-1-\zeta_{p}}dx$$ with $p=|\alpha|$, and we assume that these processes are independent for all $\alpha$. Let us arrange all the points in $\Pi_\alpha$ in the decreasing order, $$u_{\alpha 1} > u_{ \alpha 2} >\ldots >u_{\alpha n} > \ldots,
\label{us}$$ and enumerate them using the children $(\alpha n)_{n\geq 1}$ of the vertex $\alpha$. Given a vertex $\alpha\in {{\cal A}}\setminus \{*\}$ and the path $p(\alpha)$ in (\[pathtoleaf\]), we define $$w_\alpha = \prod_{\beta \in p(\alpha)} u_{\beta},
\label{ws}$$ and for the leaf vertices $\alpha \in {{\cal L}}({{\cal A}}) = {\mathbb{N}}^r$ we define $$v_\alpha = \frac{w_\alpha}{\sum_{\beta\in {\mathbb{N}}^r} w_\beta}.
\label{vs}$$ These are the weights of the Ruelle probability cascades. For other vertices $\alpha\in {{\cal A}}\setminus {{\cal L}}({{\cal A}})$ we define $$v_\alpha = \sum_{\beta\in {{\cal L}}({{\cal A}}),\,\beta\succ \alpha} v_\beta.
\label{vsall}$$ This definition obviously implies that $v_\alpha = \sum_{n\geq 1} v_{\alpha n}$ when $|\alpha|<r$. Let us now rearrange the vertex labels so that the weights indexed by children will be decreasing. For each $\alpha\in {{\cal A}}\setminus {\mathbb{N}}^r$, let $\pi_\alpha: {\mathbb{N}}\to {\mathbb{N}}$ be a bijection such that the sequence $(v_{\alpha \pi_\alpha(n)})_{n\geq 1}$ is decreasing. Using these local rearrangements we define a global bijection $\pi: {{\cal A}}\to {{\cal A}}$ in a natural way, as follows. We let $\pi(*)=*$ and then define $$\pi(\alpha n) = \pi(\alpha) \pi_{\pi(\alpha)}(n)
\label{permute}$$ recursively from the root to the leaves of the tree. Finally, we define $$V_\alpha = v_{\pi(\alpha)} \ \mbox{ for all }\ \alpha\in {{\cal A}}.
\label{Vs2}$$ The weights (\[vs\]) of the RPC will be accompanied by random fields indexed by ${\mathbb{N}}^r$ and generated along the tree ${{\cal A}}$ as follows.
**Hierarchical random fields.** Let $(\omega_\alpha)_{\alpha\in {{\cal A}}}$ be i.i.d. random variables uniform on $[0,1]$. Given a function $h: [0,1]^r \to [-1,1]$, consider a random array indexed by $\alpha= (n_1,\ldots, n_r)\in {\mathbb{N}}^r$, $$h_\alpha = h\bigl( (\omega_\beta)_{\beta\in p(\alpha)\setminus \{*\}}\bigr)
= h\bigl(\omega_{n_1},\omega_{n_1 n_2},\ldots, \omega_{n_1\ldots n_r} \bigr).
\label{MPfop}$$ Note that, especially, in subscripts or superscripts we will write $n_1\ldots n_r$ instead of $(n_1,\ldots, n_r)$. We will also denote by $(\omega_\alpha^I)_{\alpha\in {{\cal A}}}$ and $$h_\alpha^I = h\bigl( (\omega^I_\beta)_{\beta\in p(\alpha)\setminus \{*\}}\bigr)
= h\bigl(\omega^I_{n_1},\omega^I_{n_1 n_2},\ldots, \omega^I_{n_1\ldots n_r} \bigr)
\label{MPfopagain}$$ copies of the above arrays that will be independent for all different multi-indices $I.$ The function $h$ above is the second, and more complex, functional order parameter that encodes the distribution of spins inside the pure states in the Mézard-Parisi ansatz, as we shall see below. This way of generating the array $(h_\alpha)$ using a function $h: [0,1]^r \to [-1,1]$ is very redundant in a sense that there are many choices of the function $h$ that will produce the same array in distribution. However, if one prefers, there is a non-redundant (unique) way to encode an array of this type by a recursive tower of distributions on the set of distributions that is more common in physics literature.
**Extension of the definition of clause.** Let us extend the definition of function $\theta$ on $\{-1,+1\}^K$ to $[-1,+1]^K$ as follows. Often we will need to average $\exp \theta(\sigma_1,\ldots,\sigma_K)$ over $\sigma_j \in \{-1,+1\}$ (or some subset of them) independently of each other, with some weights. If we know that the average of $\sigma_j$ is equal to $x_j\in [-1,+1]$ then the corresponding measure is given by $$\mu_j({{\varepsilon}}) = \frac{1+x_j}{2} {{\rm I}}({{\varepsilon}}=1) + \frac{1-x_j}{2} {{\rm I}}({{\varepsilon}}=-1).$$ We would like to denote the average of $\exp \theta$ again by $\exp \theta$, which results in the definition $$\exp \theta(x_1,\ldots,x_K) = \sum_{\sigma_1,\ldots,\sigma_K = \pm 1} \exp \theta(\sigma_1,\ldots,\sigma_K) \prod_{j\leq K}\mu_j(\sigma_j).
\label{Deftheta}$$ Here is how this general definition would look like in the above two examples. In the first example of the $K$-spin model, using that $\sigma_1\cdots \sigma_K \in \{-1,+1\}$, we can write $$\exp \theta(\sigma_1, \ldots, \sigma_K) = {{\mbox{\rm ch}}}(\beta g)\bigl(1+\mbox{th}(\beta g)\sigma_1\cdots \sigma_K\bigr)$$ and, clearly, after averaging, $$\exp \theta(x_1,\ldots, x_K) = {{\mbox{\rm ch}}}(\beta g)\bigl(1+\mbox{th}(\beta g) x_1\cdots x_K\bigr).$$ In the second example of the $K$-sat model, using that $\prod_{j\leq K} (1+J_j \sigma_j)/2 \in\{0,1\}$, we can write $$\exp \theta(\sigma_1, \ldots, \sigma_K) =
1+(e^{-\beta}-1) \prod_{j\leq K} \frac{1+J_j \sigma_j}{2}$$ and after averaging, $$\exp \theta(x_1, \ldots, x_K) =
1+(e^{-\beta}-1) \prod_{j\leq K} \frac{1+J_j x_j}{2}.$$
**The Mézard-Parisi formula.** Let $\pi(\lambda K)$ and $\pi(\lambda(K-1))$ be Poisson random variables with the means $\lambda K$ and $\lambda (K-1)$ correspondingly and consider $$A_\alpha({{\varepsilon}})
=
\sum_{k\leq \pi( \lambda K)} \theta_{k}(h_\alpha^{1,k},\ldots,h_\alpha^{K-1,k},{{\varepsilon}})
\label{Aibef}$$ for ${{\varepsilon}}\in\{-1,+1\}$ and $$B_\alpha
=
\sum_{k\leq \pi(\lambda(K-1))} \theta_{k}(h_\alpha^{1,k},\ldots,h_\alpha^{K,k}).
\label{Bef}$$ Let ${{\rm Av}}$ denote the average over ${{\varepsilon}}=\pm 1$ and consider the following functional $${{\cal P}}(r,\zeta,h)
=
\log 2 + {\mathbb{E}}\log \sum_{\alpha\in{\mathbb{N}}^r} v_\alpha\, {{\rm Av}}\exp A_\alpha({{\varepsilon}})
-
{\mathbb{E}}\log \sum_{\alpha\in{\mathbb{N}}^r} v_\alpha \exp B_\alpha
\label{CalP}$$ that depends on $r$, the parameters (\[zetas\]) and the choice of the functions $h$ in (\[MPfop\]). Then the Mézard-Parisi ansatz predicts that $$\begin{aligned}
\lim_{N\to\infty} F_N^{\mathrm{model}} =
\inf_{r,\zeta,h} {{\cal P}}(r,\zeta,h),
\label{FE}\end{aligned}$$ at least in the above two examples of the $K$-spin and $K$-sat models. We will see below that all the parameters have a natural interpretation in terms of the structure of the Gibbs measure.
**Franz-Leone upper bound.** As we mentioned in the introduction, it was proved in [@FL] that $$\begin{aligned}
F_N^{\mathrm{model}} \leq \inf_{r,\zeta,h} {{\cal P}}(r,\zeta,h)
\label{FL}\end{aligned}$$ for all $N$, in the $K$-spin and $K$-sat models for even $K$. Their proof was rewritten in a slightly different language in [@PT] to make it technically simpler, and it was observed by Talagrand in [@SG] that the proof actually works for all $K\geq 1$ in the $K$-sat model.
As a natural starting point for proving matching lower bound, a strengthened analogue of the Aizenman-Sims-Starr representation [@AS2] for diluted models was obtained in [@Pspins] in the language of the so called asymptotic Gibbs measures. We will state this representation in Theorem \[Th1\] below for a slightly modified Hamiltonian, while also ensuring that the asymptotic Gibbs measures satisfy the Ghirlanda-Guerra identities. To state this theorem, we need to recall a few more definitions.
**Asymptotic Gibbs measures.** The Gibbs (probability) measure corresponding to a Hamiltonian $H_N(\sigma)$ on $\{-1,+1\}^N$ is defined by $$G_N(\sigma) = \frac{\exp H_N(\sigma)}{Z_N},
\label{GibbsN}$$ where the normalizing factor $Z_N=\sum_{\sigma} \exp H_N(\sigma)$ is called the partition function. To define the notion of the asymptotic Gibbs measure, we will assume that the distribution of the process $(H_N(\sigma))_{\sigma\in\{-1,+1\}^N}$ is invariant under the permutations of the coordinates of $\sigma$ - this property is called symmetry between sites, and it clearly holds in all the models we consider.
Let $(\sigma^\ell)_{\ell\geq 1}$ be an i.i.d. sequence of replicas from the Gibbs measure $G_N$ and let $\mu_N$ be the joint distribution of the array $(\sigma_i^\ell)_{1\leq i\leq N, \ell\geq 1}$ of all spins for all replicas under the average product Gibbs measure ${\mathbb{E}}G_N^{\otimes \infty}$, $$\mu_N\Bigl( \bigl\{\sigma_i^\ell = a_i^\ell \ :\ 1\leq i\leq N, 1\leq \ell \leq n \bigr\} \Bigr)
=
{\mathbb{E}}G_N^{\otimes n}\Bigl( \bigl\{\sigma_i^\ell = a_i^\ell \ :\ 1\leq i\leq N, 1\leq \ell \leq n \bigr\} \Bigr)
\label{muN}$$ for any $n\geq 1$ and any $a_i^\ell \in\{-1,+1\}$. We extend $\mu_N$ to a distribution on $\{-1,+1\}^{{\mathbb{N}}\times{\mathbb{N}}}$ simply by setting $\sigma_i^\ell=1$ for $i\geq N+1.$ Let ${{\cal M}}$ denote the set of all possible limits of $(\mu_N)$ over subsequences with respect to the weak convergence of measures on the compact product space $\{-1,+1\}^{{\mathbb{N}}\times{\mathbb{N}}}$.
Because of the symmetry between sites, all measures in ${{\cal M}}$ inherit from $\mu_N$ the invariance under the permutation of both spin and replica indices $i$ and $\ell.$ By the Aldous-Hoover representation [@Aldous], [@Hoover2] for such distributions, for any $\mu\in{{\cal M}}$, there exists a measurable function $s:[0,1]^4\to\{-1,+1\}$ such that $\mu$ is the distribution of the array $$s_i^\ell=s(w,u_\ell,v_i,x_{i,\ell}),
\label{sigma}$$ where the random variables $w,(u_\ell), (v_i), (x_{i,\ell})$ are i.i.d. uniform on $[0,1]$. The function $s$ is defined uniquely for a given $\mu\in {{\cal M}}$ up to measure-preserving transformations (Theorem 2.1 in [@Kallenberg]), so we can identify the distribution $\mu$ of array $(s_i^\ell)$ with $s$. Since $s$ takes values in $\{-1,+1\}$, the distribution $\mu$ can be encoded by the function $${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}(w,u,v) = {\mathbb{E}}_x\, s(w,u,v,x),
\label{fop}$$ where ${\mathbb{E}}_x$ is the expectation in $x$ only. The last coordinate $x_{i,\ell}$ in (\[sigma\]) is independent for all pairs $(i,\ell)$, so it plays the role of ‘flipping a coin’ with the expected value ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}(w,u_\ell,v_i)$. Therefore, given the function (\[fop\]), we can redefine $s$ by $$s(w,u_\ell,v_i,x_{i,\ell}) = 2 {{\hspace{0.3mm}{\rm I}\hspace{0.1mm}}}\Bigl(x_{i,\ell} \leq \frac{1+ {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}(w,u_\ell,v_i) }{2}\Bigr) -1
\label{sigmatos}$$ without affecting the distribution of the array $(s_i^\ell)$.
We can also view the function ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}$ in (\[fop\]) in a more geometric way as a random measure on the space of functions, as follows. Let $du$ and $dv$ denote the Lebesgue measure on $[0,1]$ and let us define a (random) probability measure $$G = G_w = du \circ \bigl(u\to {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}(w,u,\cdot)\bigr)^{-1}
\label{Gibbsw}$$ on the space of functions of $v\in [0,1]$, $$H = \bigl\{ \|{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}\|_\infty \leq 1 \bigr\},
\label{spaceH}$$ equipped with the topology of $L^2([0,1], dv)$. We will denote the scalar product in $L^2([0,1], dv)$ by $h^1\cdot h^2$ and the corresponding $L^2$ norm by $\|h\|$. The random measure $G$ in (\[Gibbsw\]) is called an [asymptotic Gibbs measure]{}. The whole process of generating spins can be broken into several steps:
1. generate the Gibbs measure $G=G_w$ using the uniform random variable $w$;
2. consider i.i.d. sequence ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\ell} = {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}(w,u_{\ell},\cdot)$ of replicas from $G$, which are functions in $H$;
3. plug in i.i.d. uniform random variables $(v_i)_{i\geq 1}$ to obtain the array ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell(v_i) = {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}(w,u_\ell,v_i)$;
4. finally, use this array to generate spins as in (\[sigmatos\]).
For a different approach to this definition via exchangeable random measures see also [@Austin]. From now on, we will keep the dependence of the random measure $G$ on $w$ implicit, denote i.i.d. replicas from $G$ by $({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell)_{\ell\geq 1}$, which are now functions on $[0,1]$, and denote the sequence of spins (\[sigmatos\]) corresponding to the replica ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell$ by $$S({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell) = \Bigl( 2 {{\hspace{0.3mm}{\rm I}\hspace{0.1mm}}}\Bigl(x_{i,\ell} \leq \frac{1+ {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell(v_i) }{2}\Bigr) -1 \Bigr)_{i\geq 1} \in \{-1,+1\}^{\mathbb{N}}.
\label{SpinsEll}$$ Because of the geometric nature of the asymptotic Gibbs measures $G$ as measures on the subset of $L^2([0,1],dv)$, the distance and scalar product between replicas play a crucial role in the description of the structure of $G$. We will denote the scalar product between replicas ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell$ and ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\ell'}$ by $R_{\ell,\ell'} = {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell\cdot {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\ell'}$, which is more commonly called [the overlap]{} of ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell$ and ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\ell'}$. Let us notice that the overlap $R_{\ell,\ell'}$ is a function of spin sequence (\[SpinsEll\]) generated by ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell$ and ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\ell'}$ since, by the strong law of large numbers, $$R_{\ell,\ell'} = \int \! {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell(v) {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\ell'}(v)\, dv =
\lim_{j\to\infty} \frac{1}{j}\sum_{i=1}^j S\bigl({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell\bigr)_i \,S\bigl({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\ell'}\bigr)_i
\label{overlapinfty}$$ almost surely.
**The Ghirlanda-Guerra identities.** Given $n\geq 1$ and replicas ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^1,\ldots, {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^n$, we will denote the array of spins (\[SpinsEll\]) corresponding to these replicas by $$S^n = \bigl(S({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell) \bigr)_{1\leq \ell\leq n}.
\label{Sn}$$ We will denote by ${\langle}\,\cdot\,{\rangle}$ the average over replicas ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell$ with respect to $G^{\otimes \infty}$. In the interpretation of the step (ii) above, this is the same as averaging over $(u_\ell)_{\ell\geq 1}$ in the sequence $({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}(w,u_{\ell},\cdot))_{\ell\geq 1}$. Let us denote by ${\mathbb{E}}$ the expectation with respect to random variables $w$, $(v_i)$ and $x_{i,\ell}$.
We will say that the measure $G$ on $H$ satisfies the Ghirlanda-Guerra identities if for any $n\geq 2,$ any bounded measurable function $f$ of the spins $S^n$ in (\[Sn\]) and any bounded measurable function $\psi$ of one overlap, $${\mathbb{E}}\bigl{\langle}f(S^n)\psi(R_{1,n+1}) \bigr{\rangle}= \frac{1}{n}\hspace{0.3mm} {\mathbb{E}}\bigl{\langle}f(S^n) \bigr{\rangle}\hspace{0.3mm} {\mathbb{E}}\bigr{\langle}\psi(R_{1,2})\bigr{\rangle}+ \frac{1}{n}\sum_{\ell=2}^{n}{\mathbb{E}}\bigl{\langle}f(S^n)\psi(R_{1,\ell})\bigr{\rangle}.
\label{GG}$$ Another way to express the Ghirlanda-Guerra identities is to say that, conditionally on $S^n$, the law of $R_{1,n+1}$ is given by the mixture $$\frac{1}{n} \hspace{0.3mm}\zeta + \frac{1}{n}\hspace{0.3mm} \sum_{\ell=2}^n \delta_{R_{1,\ell}},
\label{GGgen}$$ where $\zeta$ denotes the distribution of $R_{1,2}$ under the measure ${\mathbb{E}}G^{\otimes 2}$, $$\zeta(\ \cdot\ ) = {\mathbb{E}}G^{\otimes 2}\bigl(R_{1,2}\in\ \cdot\ \bigr).
\label{zeta}$$ The identities (\[GG\]) are usually proved for the function $f$ of the overlaps $(R_{\ell,\ell'})_{\ell,\ell'\leq n}$ instead of $S^n$, but exactly the same proof yields (\[GG\]) as well (see e.g. Section 3.2 in [@SKmodel]). It is well known that these identities arise from the Gaussian integration by parts of a certain Gaussian perturbation Hamiltonian against the test function $f$, and one is free to choose this function to depend on all spins and not only overlaps.
**Modification of the model Hamiltonian.** Next, we will describe a crucial new ingredient that will help us classify all finite-RSB asymptotic Gibbs measures. Let us consider a sequence $(g^d)_{d\geq 1}$ of independent Gaussian random variables satisfying $${\mathbb{E}}(g^d)^2 \leq 2^{-d}\epsilon^{\mathrm{pert}},
\label{gsvar}$$ where ${{\varepsilon}}^{\mathrm{pert}}$ is a fixed small parameter, and consider the following random clauses of $d$ variables, $$\theta^d(\sigma_1,\ldots,\sigma_d) = g^d \frac{1+\sigma_{1}}{2}\cdots \frac{1+\sigma_{d}}{2}.$$ We will denote by $g_{I}^d$ and $\theta_{I}^d$ independent copies over different multi-indices $I$. We will define a perturbation Hamiltonian by $$H_N^{\mathrm{pert}}(\sigma)=\sum_{i\leq N} \theta_{i}^1(\sigma_i)
+\sum_{d\geq 2} \sum_{k\leq \pi_d(N)}\theta_{k}^d(\sigma_{i_{1,d,k}},\ldots, \sigma_{i_{d,d,k}}),
\label{HNpertmain}$$ where $\pi_d(N)$ are Poisson random variables with the mean $N$ independent over $d\geq 2$ and $i_{I}$ are chosen uniformly from $\{1,\ldots, N\}$ independently for different indices $I$. Notice that, because of (\[gsvar\]), this Hamiltonian is well defined. We will now work with the new Hamiltonian given by $$H_N(\sigma) = H_N^{\mathrm{model}}(\sigma) + H_N^{\mathrm{pert}}(\sigma).
\label{HamMain}$$ Notice that the second term $H_N^{\mathrm{pert}}$ is of the same order as the model Hamiltonian, but its size is controlled by the parameter ${{\varepsilon}}^{\mathrm{pert}}$. For example, if we consider the free energy $$F_N = \frac{1}{N}{\mathbb{E}}\log \sum_{\sigma\in \Sigma_N} \exp H_N(\sigma)
\label{FNmod}$$ corresponding to this modified Hamiltonian, letting ${{\varepsilon}}^\mathrm{pert}$ go to zero will give the free energy of the original model.
Finally, as in (\[Deftheta\]), let us extend the definition of $\theta^d$ by $$\exp \theta^d(x_1,\ldots,x_d) = 1+(e^{g^d}-1) \frac{1+x_{1}}{2}\cdots \frac{1+x_{d}}{2}
\label{thetadetx}$$ to $[-1,+1]^d$ from $\{-1,+1\}^d.$
**The cavity equations for the modified Hamiltonian.** Let us now recall the cavity equations for the distribution of spins proved in [@Pspins]. These equations will be slightly modified here to take into account that the perturbation Hamiltonian $H_N^{\mathrm{pert}}(\sigma)$ will now also contribute to the cavity fields.
We will need to pick various sets of different spin coordinates in the array $(s_i^\ell)$ in (\[sigma\]), and it is inconvenient to enumerate them using one index $i\geq 1$. Instead, we will use multi-indices $I= (i_1,\ldots, i_n)$ for $n\geq 1$ and $i_1,\ldots, i_n\geq 1$ and consider $$s_{I}^\ell = s(w,u_\ell, v_{I},x_{I,\ell}),$$ where all the coordinates are uniform on $[0,1]$ and independent over different sets of indices. Similarly, we will denote $${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{I}^\ell = {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\ell}(v_I) = {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}(w,u_\ell, v_{I}).$$ For convenience, below we will separate averaging over different replicas $\ell$, so when we average over one replica we will drop the superscript $\ell$ and simply write $${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{I} = {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}(v_I) = {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}(w,u, v_{I}).
\label{sG}$$ Now, take arbitrary integers $n, m, q\geq 1$ such that $n\leq m.$ The index $q$ will represent the number of replicas selected, $m$ will be the total number of spin coordinates and $n$ will be the number of cavity coordinates. For each replica index $\ell\leq q$ we consider an arbitrary subset of coordinates $C_\ell\subseteq \{1,\ldots, m\}$ and split them into cavity and non-cavity coordinates, $$C_\ell^1 = C_\ell\cap \{1,\ldots, n\},\,\,\,
C_\ell^2=C_\ell\cap \{n+1,\ldots,m\}.
\label{C12}$$ The following quantities represent the $i$th coordinate cavity field of the modified Hamiltonian (\[HamMain\]) in the thermodynamic limit, $$\begin{aligned}
A_i({{\varepsilon}}) =& \sum_{k\leq \pi_i(\lambda K)} \theta_{k,i}( {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{1,k,i}, \ldots, {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{K-1,k,i}, {{\varepsilon}})
\nonumber
\\
& + \theta_i^1({{\varepsilon}})
+\sum_{d\geq 2} \sum_{k\leq \pi_i(d)}\theta_{k,i}^d({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{{1,d,k,i}},\ldots, {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{{d-1,d,k,i}},{{\varepsilon}}),
\label{Ai}\end{aligned}$$ where $\pi_i(d)$ and $\pi_i(\lambda K)$ are Poisson random variables with the mean $d$ and $\lambda K$, independent of each other and independent over $d\geq 2$ and $i\geq 1$. Compared to [@Pspins], now we have additional terms in the second line in (\[Ai\]) coming from the perturbation Hamiltonian (\[HNpertmain\]). Next, let us denote $${A}_i = \log {{\rm Av}}\exp {A}_i({{\varepsilon}}) \ \mbox{ and }\
\xi_i = \frac{{{\rm Av}}{{\varepsilon}}\exp {A}_i({{\varepsilon}}) }{\exp {A}_i},$$ where ${{\rm Av}}$ denotes the uniform average over ${{\varepsilon}}= \pm 1$. Recall that ${\langle}\,\cdot\,{\rangle}$ denotes the average with respect to the asymptotic Gibbs measure $G$. Define $${U}_\ell = \Bigl{\langle}\prod_{i\in C_\ell^1} \xi_i \prod_{i\in C_\ell^2} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i
\,\exp \sum_{i\leq n} {A}_i \Bigr{\rangle}\ \mbox{ and } \
{V} =\Bigl{\langle}\exp \sum_{i\leq n} {A}_i \Bigr{\rangle}.
\label{Ulbar2}$$ Then we will say that an asymptotic Gibbs measure $G$ satisfies the cavity equations if $${\mathbb{E}}\prod_{\ell\leq q} \Bigl{\langle}\, \prod_{i\in C_\ell} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i \Bigr{\rangle}={\mathbb{E}}\prod_{\ell\leq q} \frac{U_\ell}{V}
\label{SC}$$ for all choice of $n,m,q$ and sets $C_\ell^1, C_\ell^2$.
**The Aizenman-Sims-Starr type lower bound.** Consider a random measure $G$ on $H$ in (\[spaceH\]) and let ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_I$ be generated by a replica ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}$ from this measure as in (\[sG\]). From now on we will denote by $\pi(c)$ a Poisson random variable with the mean $c$ and we will assume that different appearances of these in the same equation are independent of each other and all other random variables. This means that if we write $\pi(a)$ and $\pi(b)$, we assume them to be independent even if $a$ happens to be equal to $b$. Consider $$\begin{aligned}
A({{\varepsilon}}) =& \sum_{k\leq \pi(\lambda K)} \theta_{k}( {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{1,k}, \ldots, {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{K-1,k}, {{\varepsilon}})
\nonumber
\\
& + \theta_i^1({{\varepsilon}})
+\sum_{d\geq 2} \sum_{k\leq \pi(d)}\theta_{k}^d({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{{1,d,k}},\ldots, {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{{d-1,d,k}},{{\varepsilon}}),
\label{Aiag}\end{aligned}$$ for ${{\varepsilon}}\in\{-1,+1\}$ and $$\begin{aligned}
B =& \sum_{k\leq \pi(\lambda (K-1))} \theta_{k}( {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{1,k}, \ldots, {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{K,k})
\nonumber
\\
& +\sum_{d\geq 2} \sum_{k\leq \pi(d-1)}\theta_{k}^d({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{{1,d,k}},\ldots, {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{{d,d,k}}).
\label{Bag}\end{aligned}$$ Again, compared to [@Pspins], we have additional terms in the second line in (\[Aiag\]) and (\[Bag\]) coming from the perturbation Hamiltonian (\[HNpertmain\]). Consider the following functional $${{\cal P}}(G)
=
\log 2 + {\mathbb{E}}\log \Bigl{\langle}{{\rm Av}}\exp A({{\varepsilon}}) \Bigr{\rangle}-
{\mathbb{E}}\log \Bigl{\langle}\exp B \Bigr{\rangle}.
\label{PPG}$$ The following is a slight modification of the (lower bound part of the) main result in [@Pspins] in the setting of the diluted models.
\[Th1\] The lower limit of the free energy in (\[FNmod\]) satisfies $$\liminf_{N\to \infty} F_N \geq \inf_G {{\cal P}}(G),
\label{PPGeq}$$ where the infimum is taken over random measures $G$ on $H$ that satisfy the Ghirlanda-Guerra identities (\[GG\]) and the cavity equations (\[SC\]).
We will call the measures $G$ that appear in this theorem asymptotic Gibbs measures, because that is exactly how they arise in [@Pspins]. The main difference from [@Pspins] is that we also include the requirement that the measures $G$ satisfy the Ghirlanda-Guerra identities in addition to the cavity equations. This can be ensured in exactly the same way as in the Sherrington-Kirkpatrick model by way of another small perturbation of the Hamiltonian (see e.g. [@HEPS], where this was explained for the $K$-sat model). We are not going to prove Theorem \[Th1\] in this paper, because it does not require any new ideas which are not already explained in [@Pspins; @HEPS; @1RSB], and the main reason we stated it here is to provide the motivation for our main result below. Of course, the proof involves some technical modifications to take into account the presence of the new perturbation term (\[HNpertmain\]), but these are not difficult.
Instead, we will focus on the main new idea and the main new contribution of the paper, which is describing the structure of measures $G$ that satisfy the Ghirlanda-Guerra identities and the cavity equations in the case when the overlap $R_{1,2} = {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}\cdot {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}'$ of any two points ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}$ and ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}'$ in the support of $G$ takes finitely many, say, $r+1$ values, $$0\leq q_0< q_1<\ldots< q_r \leq 1,
\label{finiteoverlap}$$ for any $r\geq 1$ - the so called $r$-step replica symmetry breaking (or $r$-RSB) case. To state the main result, let us first recall several known consequences of the Ghirlanda-Guerra identities.
**Consequences of the Ghirlanda-Guerra identities.** By Talagrand’s positivity principle (see [@SG; @SKmodel]), if the Ghirlanda-Guerra identities hold then the overlap can take only non-negative values, so the fact that the values in (\[finiteoverlap\]) are between $0$ and $1$ is not a constraint. Another consequence of the Ghirlanda-Guerra identities (Theorem 2.15 in [@SKmodel]) is that with probability one the random measure $G$ is concentrated on the sphere on radius $\sqrt{q_r}$, i.e. $G(\|{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}\|^2 = q_r)=1.$ Since we assume that the overlap takes finitely many values, $G$ is also purely atomic. Finally (see [@PUltra] or Theorem 2.14 in [@SKmodel]), with probability one the support of $G$ is ultrametric, $
G^{\otimes 3}(R_{2,3} \geq \min(R_{1,2},R_{1,3}))=1.
$ By ultrametricity, for any $q_p$, the relation defined by $${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}\sim_{q_p} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}' \Longleftrightarrow {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}\cdot{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}' \geq q_p
\label{qclusters}$$ is an equivalence relation on the support of $G$. We will call these $\sim_q$ equivalence clusters simply $q$-clusters. Let us now enumerate all the $q_p$-clusters defined by (\[qclusters\]) according to Gibbs’ weights as follows. Let $H_{*}$ be the entire support of $G$ so that $V_* = G(H_*) =1$. Next, the support is split into $q_1$-clusters $(H_n)_{n\geq 1}$, which are then enumerated in the decreasing order of their weights $V_n = G(H_n)$, $$V_1 > V_2 > \ldots > V_n > \ldots > 0.
\label{purelabelsfirst}$$ We then continue recursively over $p\leq r-1$ and enumerate the $q_{p+1}$-subclusters $(H_{\alpha n})_{n\geq 1}$ of a cluster $H_\alpha$ for $\alpha\in {\mathbb{N}}^p$ in the non-increasing order of their weights $V_{\alpha n} = G(H_{\alpha n})$, $$V_{\alpha 1} > V_{\alpha 2} > \ldots > V_{\alpha n} > \ldots > 0.
\label{purelabels}$$ Thus, all these clusters were enumerated $(V_\alpha)_{\alpha\in {{\cal A}}}$ by the vertices of the tree ${{\cal A}}$ in (\[Atree\]). It is not a coincidence that we used the same notation as in (\[Vs2\]). It is another well-known consequence of the Ghirlanda-Guerra identities that the distribution of these weights coincides with the reordering of the weights of the Ruelle probability cascades as in (\[Vs2\]) with the parameters (\[zetas\]) given by $$\zeta_p = {\mathbb{E}}G^{\otimes 2}\bigl( R_{1,2} \leq q_p \bigr)
\label{zetap}$$ for $p=0,\ldots,r.$ The $q_r$-clusters are the points of the support of $G$ - these are called pure states. They were enumerated by $\alpha\in{\mathbb{N}}^r$ and, if we denote them by ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_\alpha$, $$G({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_\alpha) = V_\alpha \,\,\mbox{ for }\,\, \alpha\in {\mathbb{N}}^r.
\label{Gdiscrete}$$ Recall that we generate the array ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\ell$ (or ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_I^\ell$ for general index $I$) by first sampling replicas ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell$ from the measure $G$ (which are functions on $[0,1]$) and then plugging in i.i.d. uniform random variables $v_i$, i.e. ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\ell = {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell(v_i)$. In the discrete setting (\[Gdiscrete\]), this is equivalent to sampling $\alpha$ according to the weights $V_\alpha$ and then plugging in $v_i$ into ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_\alpha,$ i.e. ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_\alpha(v_i)$. Therefore, in order to describe the distribution of the array $({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\ell(v_i))_{i,\ell\geq 1}$ it is sufficient to describe the joint distribution of the arrays $$\bigl(V_{\alpha}\bigr)_{\alpha\in{\mathbb{N}}^r} \,\,\mbox{ and }\,\,
\bigl({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{\alpha}(v_i)\bigr)_{i\geq 1, \alpha\in{\mathbb{N}}^r}.$$ In addition to the fact that $(V_\alpha)$ corresponds to some reordering of weights of the Ruelle probability cascades, it was proved in [@AP; @HEPS] that if the measure $G$ satisfies the Ghirlanda-Guerra identities then (see Theorem $1$ and equation (36) and (37) in [@HEPS]):
1. the arrays $(V_{\alpha})_{\alpha\in{\mathbb{N}}^r}$ and $({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{\alpha}(v_i))_{i\geq 1, \alpha\in{\mathbb{N}}^r}$ are independent;
2. there exists a function $h:[0,1]^{2(r+1)}\to[-1,1]$ such that $$\Bigl({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_\alpha(v_i) \Bigr)_{i\geq 1, \alpha\in{\mathbb{N}}^r}
\, \stackrel{d}{=}\,
\Bigl( h\bigl( (\omega_\beta)_{\beta\in p(\alpha)}, (\omega_\beta^i)_{\beta\in p(\alpha)} \bigr) \Bigr)_{i\geq 1,\alpha\in{\mathbb{N}}^r},
\label{sigmaf2}$$ where, as above, $\omega_\alpha$ and $\omega_\alpha^i$ for $\alpha\in{{\cal A}}$ are i.i.d. uniform random variables on $[0,1]$.
The Mézard-Parisi ansatz predicts that in the equation (\[sigmaf2\]) one can replace the function $h$ by a function that does not depend on the coordinates $(\omega_\beta)_{\beta\in p(\alpha)}$, which would produce exactly the same fields as in (\[MPfopagain\]). We will show that this essentially holds for finite-RSB asymptotic Gibbs measures.
**Consequence of the cavity equations.** The main result of the paper is the following.
\[Th2\] If a random measure $G$ on $H$ in (\[spaceH\]) satisfies the Ghirlanda-Guerra identities (\[GG\]) and the cavity equations (\[SC\]) and the overlap takes $r+1$ values in (\[finiteoverlap\]) then there exists a function $h:[0,1]^{r+2}\to[-1,1]$ such that $$\Bigl({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_\alpha(v_i) \Bigr)_{i\geq 1, \alpha\in{\mathbb{N}}^r}
\, \stackrel{d}{=}\,
\Bigl( h\bigl( \omega_*, (\omega_\beta^i)_{\beta\in p(\alpha)} \bigr) \Bigr)_{i\geq 1,\alpha\in{\mathbb{N}}^r},
\label{sigmaf3}$$ where $\omega_*$ and $\omega_\alpha^i$ for $\alpha\in{{\cal A}}$ are i.i.d. uniform random variables on $[0,1]$.
In other words, the cavity equations (\[SC\]) allow us to simplify (\[sigmaf2\]) and to get rid of the dependence on the coordinates $\omega_\beta$ for $\beta\in {{\cal A}}\setminus \{*\}$. Notice that, compared to the Mézard-Parisi ansatz, we still have the dependence on $\omega_*$ in (\[sigmaf3\]). However, from the point of view of computing the free energy this is not an issue at all, because the average in $\omega_*$ is on the outside of the logarithm in (\[PPG\]) and when we minimize over $G$ in (\[PPGeq\]), we can replace the average over $\omega_*$ by the infimum. Of course, the infimum over $G$ in (\[PPGeq\]) could involve measures that are not of finite-RSB type, and this is the main obstacle to finish the proof of the Mézard-Parisi formula, if this approach can be made to work.
If one could replace the infimum in (\[PPGeq\]) over measures $G$ that satisfy the finite-RSB condition in (\[finiteoverlap\]) (in addition to the cavity equations and the Ghirlanda-Guerra identities) then, using Theorem \[Th2\] and replacing the average over $\omega_*$ by the infimum, we get the lower bound that essentially matches the Franz-Leone upper bound, except that now we have additional terms in the second line in (\[Aiag\]) and (\[Bag\]) compared to (\[Aibef\]) and (\[Bef\]) coming from the perturbation Hamiltonian (\[HNpertmain\]). However, these terms are controlled by ${{\varepsilon}}^\mathrm{pert}$ in (\[gsvar\]) and, letting it go to zero, one could remove the dependence of the lower bound on these terms and match the Franz-Leone upper bound.
General idea of the proof {#Sec2ilabel}
=========================
The main goal of this paper is to show that the function $h$ that generates the array ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\alpha$ in (\[sigmaf2\]), $${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\alpha = h\bigl( (\omega_\beta)_{\beta\in p(\alpha)}, (\omega_\beta^i)_{\beta\in p(\alpha)} \bigr),$$ can be replaced by a function that does not depend on the coordinates $\omega_\beta$ for $*\prec \beta\preceq \alpha$. We will show this by induction, removing one coordinate at a time from the leaf $\alpha$ up to the root $*$. Our induction assumption will be the following: for $p\in \{0,\ldots, r-1\}$, suppose that, instead of (\[sigmaf2\]), the array ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\alpha$ for $i\geq 1,\alpha\in {\mathbb{N}}^r$, is generated by $${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\alpha = h\bigl( (\omega_\beta)_{\beta\in p(\alpha), |\beta|\leq p+1}, (\omega_\beta^i)_{\beta\in p(\alpha)} \bigr)
\label{indass}$$ for some function $h$ that does not depend on the coordinates $\omega_\beta$ for $|\beta|\geq p+2$. Notice that this holds for $p=r-1$, and we would like to show that one can replace $h$ by $h'$ that also does not depend on $\omega_{\beta}$ for $|\beta|=p+1$, without affecting the distribution of the array ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\alpha$.
Often, we will work with a subtree of ${{\cal A}}$ that ‘grows out’ of a vertex at the distance $p$ or $p+1$ from the root, which means that all paths from the root to the vertices in that subtree pass through this vertex. In that case, for certainty, we will fix the vertex to be $[p] = (1,2,\ldots,p)$ or $[p+1]$. We will denote by ${{\mathbb{E}_{[p]}}}$ the expectation with respect to the random variables $\omega_\beta$, $\omega_\beta^i$ indexed by the descendants of $[p]$, i.e. $[p]\prec \beta$, and by ${{\mathbb{E}_{[p],i}}}$ the expectation with respect to $\omega_\beta^i$ for $[p]\prec \beta$. Our goal will be to prove the following.
\[Sec6iTh1\] Under the assumption (\[indass\]), for any $k\geq 1$ and any $[p+1]\preceq \alpha_1,\ldots,\alpha_k \in {\mathbb{N}}^r$, the expectation ${{\mathbb{E}_{[p],i}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_1}\cdots {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_k}$ with respect to $(\omega_\beta^i)_{[p+1]\preceq \beta}$ does not depend on $\omega_{[p+1]}$ almost surely.
Here $i\geq 1$ is arbitrary but fixed and $\alpha_1,\ldots,\alpha_k$ need not be different, so the quantities ${{\mathbb{E}_{[p],i}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_1}\cdots {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_k}$ represent all possible joint moments with respect to $(\omega_\beta^i)_{[p+1]\preceq \beta}$ of the random variables ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\alpha$ for $[p+1]\preceq \alpha\in {\mathbb{N}}^r$. It will take us the rest of the paper to prove this result, and right now we will only explain why it completes the induction step.
The reason is identical to the situation of an exchangeable sequence $X_n = h(\omega,\omega_n)$ (say, bounded in absolute value by one) such that all moments ${\mathbb{E}}_{\omega_1} X_1^k$ for $k\geq 1$ with respect to $\omega_1$ do not depend on $\omega$. In this case if we choose any function $h'(\omega_1)$ with this common set of moments then the sequences $(h'(\omega_n))$ and $(X_n)$ have the same distribution, which can be seen by comparing their joint moments. For example, we can choose $h'(\,\cdot\,) = h(\omega^*,\,\cdot\,)$ for any $\omega^*$ from the set of measure one on which all moments ${\mathbb{E}}_{\omega_1} X_1^k$ coincide with their average values ${\mathbb{E}}X_1^k$.
We can do the same in the setting of Theorem \[Sec6iTh1\], which can be rephrased as follows: for almost all $(\omega_\beta,\omega_\beta^i)_{\beta\preceq [p]}$ and $\omega_{[p+1]}$, $${{\mathbb{E}_{[p],i}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_1}\cdots {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_k} = {{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_1}\cdots {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_k}$$ for all $k\geq 1$ and $[p+1]\preceq \alpha_1,\ldots,\alpha_k \in {\mathbb{N}}^r$, where ${{\mathbb{E}_{[p]}}}$ now also includes the average in $\omega_{[p+1]}.$ This means that we can find $\omega_{[p+1]}=\omega_{[p+1]}^*$ such that the equality of all these moments holds for almost all $(\omega_\beta,\omega_\beta^i)_{\beta\preceq [p]}$. If we now set $$h'\bigl( (\omega_\beta)_{\beta\in p(\alpha), |\beta|\leq p}, (\omega_\beta^i)_{\beta\in p(\alpha)} \bigr)
=
h\bigl( (\omega_\beta)_{\beta\in p(\alpha), |\beta|\leq p}, \omega_{[p+1]}^*, (\omega_\beta^i)_{\beta\in p(\alpha)} \bigr)$$ then by comparing the joint moments one can see that $$\bigl({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\alpha\bigr)_{i\geq 1,\alpha\in{\mathbb{N}}^r} \stackrel{d}{=}
\Bigl(h'\bigl( (\omega_\beta)_{\beta\in p(\alpha), |\beta|\leq p}, (\omega_\beta^i)_{\beta\in p(\alpha)} \bigr) \Bigr)_{i\geq 1,\alpha\in{\mathbb{N}}^r},$$ which completes the induction step.
The proof of Theorem \[Sec6iTh1\] will proceed by a certain induction on the shape of the configuration $\alpha_1,\ldots,\alpha_k$, where by the shape of the configuration we essentially mean the matrix $(\alpha_\ell\wedge \alpha_{\ell'})_{\ell,\ell'\leq k}$ (or its representation by a tree that consists of all paths $p(\alpha_\ell)$). It is clear that the quantity $${{\mathbb{E}_{[p],i}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_1}\cdots {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_k}$$ depends on $[p+1]\preceq \alpha_1,\ldots,\alpha_k \in {\mathbb{N}}^r$ only through the shape. The induction will be somewhat involved and we will explain exactly how it will work toward the end of the paper, once we have all the tools ready. However, we need to mention now that the induction will have an important *monotonicity property*: whenever we have proved the statement of Theorem \[Sec6iTh1\] for some $\alpha_1,\ldots,\alpha_k$, we have also proved it for any subset of these vertices. At this moment, we will suppose that $[p+1]\preceq \alpha_1,\ldots,\alpha_k \in {\mathbb{N}}^r$ are such that the following holds:
1. For any subset $S\subseteq \{1,\ldots, k\}$, ${{\mathbb{E}_{[p],i}}}\prod_{\ell\in S}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}$ does not depend on $\omega_{[p+1]}$.
Then over the next sections we will obtain some implications of this assumption using the cavity equations. Finally, in the last section we will show how to use these implications inductively to prove Theorem \[Sec6iTh1\] for any choice of $\alpha_1,\ldots,\alpha_k$. Of course, the starting point of the induction will be the case of $k=1$ that we will obtain first. In fact, in this case the statement will be even stronger and will not assume that (\[indass\]) holds (i.e. we only assume (\[sigmaf2\])).
\[Sec2iLem1\] For any $p=0,\ldots, r-1$ and any $[p]\prec \alpha \in {\mathbb{N}}^r$, the expectation ${{\mathbb{E}_{[p],i}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha}$ with respect to $(\omega_\beta^i)_{[p]\prec \beta}$ does not depend on $(\omega_\beta)_{[p]\prec \beta}$ almost surely.
**Proof.** Consider $[p]\prec \alpha, \beta\in{\mathbb{N}}^r$ such that $\alpha\wedge \beta = p$. By (\[sigmaf2\]), it is clear that the overlap of two pure states satisfies $$R_{\alpha, \beta}:= {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_\alpha\cdot {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{\beta}
=
\int_0^1\! {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_\alpha(v) {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_{\beta}(v) \, dv
\stackrel{d}{=}
{\mathbb{E}}_i \, {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\beta},
\label{Sec2eq1}$$ where ${\mathbb{E}}_i$ denotes the expectation in random variables $\omega_\eta^i$ that depend on the spin index $i$. By construction, we enumerated the pure states ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_\alpha$ in (\[Gdiscrete\]) so that $$R_{\alpha,\beta} = q_{\alpha\wedge\beta}
\label{RabSec2}$$ and, since $\alpha\wedge \beta = p$, we get that, almost surely, $$q_p = R_{\alpha,\beta} = {\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\beta}
= {\mathbb{E}}_i \bigl({{\mathbb{E}_{[p],i}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha} {{\mathbb{E}_{[p],i}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\beta}\bigr).$$ If we denote $v = (\omega_\eta)_{\eta\preceq [p]}$, $v_1 = (\omega_\eta)_{[p]\prec \eta\preceq \alpha}$, $v_2 = (\omega_\eta)_{[p]\prec \eta\preceq \beta}$ and $u = (\omega_\eta^i)_{\eta\preceq [p]}$ then $${{\mathbb{E}_{[p],i}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha} =\varphi(v,v_1,u),\,\,
{{\mathbb{E}_{[p],i}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\beta}=\varphi(v,v_2,u)$$ for some function $\varphi,$ the random variables $v,v_1,v_2,u$ are independent, and the above equation can be written as $$q_p = {\mathbb{E}}_u \varphi(v,v_1,u) \varphi(v,v_2,u)$$ for almost all $v,v_1,v_2$. This means that for almost all $v$, the above equality holds for almost all $v_1, v_2$. Let us fix any such $v$ and let $\mu_v$ be the image of the Lebesgue measure on $[0,1]^{r-p}$ by the map $v_1 \to \varphi(v,v_1,\cdot)\in L^2([0,1]^{p+1},du)$. Then the above equation means that, if we sample independently two points from $\mu_v$, with probability one their scalar product in $L^2$ will be equal to $q_p$. This can happen only if the measure $\mu_v$ is concentrated on one point in $L^2$, which means that the function $\varphi$ does not depend on $v_1$.
Before we start using the cavity equations, we will explain a property of the Ruelle probability cascades that will play the role of the main technical tool throughout the paper.
Key properties of the Ruelle probability cascades {#Sec2label}
=================================================
The property described in this section will be used in two ways - directly, in order to obtain some consequences of the cavity equations, and indirectly, as a representation tool to make certain computations possible. This property is proved in Theorem 4.4 in [@SKmodel] in a more general form, but here we will need only a special case as follows.
Let us consider a random variable $z$ taking values in some measurable space $({{\cal X}}, {\cal B})$ (in our case, this will always be some nice space, such as $[0,1]^n$ with the Borel $\sigma$-algebra) and let $z_\alpha$ be its independent copies indexed by the vertices of the tree $\alpha\in {{\cal A}}\setminus \{*\}$ excluding the root. Recall the parameters $(\zeta_p)_{0\leq p\leq r-1}$ in (\[zetas\]). Let us consider a measurable bounded function $$X_r:{{\cal X}}^r \to {\mathbb{R}}\label{Xr}$$ and, recursively over $0\leq p\leq r-1$, define functions $X_p:{{\cal X}}^p \to{\mathbb{R}}$ by $$X_p(x) = \frac{1}{\zeta_p}\log {\mathbb{E}}_z \exp \zeta_p X_{p+1}(x,z),
\label{ch54Xp}$$ where the expected value ${\mathbb{E}}_z$ is with respect to $z$. In particular, $X_0$ is a constant. Let us define $$W_p(x,y) = \exp \zeta_p\bigl(X_{p+1}(x,y) - X_p(x)\bigr)
\label{ch31Wp}$$ for $x\in {{\cal X}}^p$ and $y\in{{\cal X}}.$ Let us point out that, by the definition (\[ch54Xp\]), ${\mathbb{E}}_z W_p(x,z) = 1$ and, therefore, for each $x\in{{\cal X}}^p,$ we can think of $W_p(x,\, \cdot\, )$ as a change of density that yields the following conditional distribution on ${{\cal X}}$ given $x\in{{\cal X}}^p$, $$\nu_p(x,B) = {\mathbb{E}}_z W_p(x,z)\, {{\rm I}}(z\in B).
\label{ch51transitionprime}$$ For $p=0$, $\nu_0$ is just a probability distribution on $({{\cal X}},{\cal B}).$ Let us now generate the array $\tilde{z}_\alpha$ for $\alpha\in {{\cal A}}\setminus \{*\}$ iteratively from the root to the leaves as follows. Let $\tilde{z}_n$ for $n\in{\mathbb{N}}$ be i.i.d. random variables with the distribution $\nu_0$. If we already constructed $\tilde{z}_\alpha$ for $|\alpha|\leq p$ then, given any $\alpha\in {\mathbb{N}}^p$, we generate $\tilde{z}_{\alpha n}$ independently for $n\geq 1$ from the conditional distribution $\nu_p((z_\beta)_{*\prec \beta\preceq \alpha},\, \cdot\,)$, and these are generated independently over different such $\alpha.$ Notice that the distribution of the array $(\tilde{z}_\alpha)_{\alpha\in{{\cal A}}\setminus\{*\}}$ depends on the distribution of $z$, function $X_r$ and parameters $(\zeta_p)_{0\leq p\leq r-1}$.
With this definition, the expectation ${{\mathbb{E}}} f((\tilde{z}_\alpha)_{\alpha\in F})$ for a finite subset $F\subset {{\cal A}}\setminus\{*\}$ can be written as follow. For $\alpha\in{{\cal A}}\setminus \{*\}$, let $$W_\alpha = W_{|\alpha|-1}\bigl( (z_\beta)_{*\prec \beta\prec \alpha}, z_\alpha \bigr).
\label{Sec2Walpha}$$ Slightly abusing notation, we could also write this simply as $W_\alpha = W_{|\alpha|-1}( (z_\beta)_{*\prec \beta\preceq \alpha})$. Given a finite subset ${C} \subset {{\cal A}}\setminus\{*\}$, let $$p({C}) = \bigcup_{\alpha\in {C}} p(\alpha) \setminus \{*\}
\,\,\mbox{ and }\,\,
W_{{C}} = \prod_{\alpha\in p({C})} W_\alpha.$$ Then, the above definition of the array $(\tilde{z}_\alpha)$ means that $${{\mathbb{E}}} f\bigl((\tilde{z}_\alpha)_{\alpha\in {C}} \bigr)
=
{\mathbb{E}}W_{{C}} f\bigl(({z}_\alpha)_{\alpha\in {C}}\bigr).
\label{Sec2expectW}$$ Simply, to average over $(\tilde{z}_\alpha)_{\alpha\in {C}}$ we need to use changes of density over all vertices in the paths from the root leading to the vertices $\alpha\in {C}$.
The meaning of the above construction will be explained by the following result. Recall the Ruelle probability $(v_\alpha)_{\alpha\in {\mathbb{N}}^r}$ cascades in (\[vs\]) and define new random weights on ${\mathbb{N}}^r$, $$\tilde{v}_\alpha
=
\frac{v_\alpha \exp X_r((z_\beta)_{*\prec \beta\preceq \alpha})}{\sum_{\alpha\in {\mathbb{N}}^r}v_\alpha \exp X_r((z_\beta)_{*\prec \beta\preceq \alpha})},
\label{tildevs}$$ by the change of density proportional to $\exp X_r((z_\beta)_{*\prec \beta\preceq \alpha})$. We will say that a bijection $\pi:{{\cal A}}\to{{\cal A}}$ of the vertices of the tree ${{\cal A}}$ preserves the parent-child relationship if children $\alpha n$ of $\alpha$ are mapped into children of $\pi(\alpha)$, $\pi(\alpha n) = (\pi(\alpha),k)$ for some $k\in{\mathbb{N}}$. Another way to write this is to say that $\pi(\alpha)\wedge \pi(\beta) = \alpha\wedge\beta$ for all $\alpha,\beta\in {{\cal A}}.$ For example, the bijection $\pi$ defined in (\[permute\]), (\[Vs2\]), is of this type. Theorem 4.4 in [@SKmodel] gives the following generalization of the Bolthausen-Sznitman invariance property for the Poisson-Dirichlet point process (Proposition A.2 in [@Bolthausen]).
There exists a random bijection $\rho:{{\cal A}}\to{{\cal A}}$ of the vertices of the tree ${{\cal A}}$, which preserves the parent-child relationship, such that $$\bigl(\tilde{v}_{\rho(\alpha)}\bigr)_{\alpha\in{\mathbb{N}}^r}
\stackrel{d}{=}
\bigl(v_\alpha\bigr)_{\alpha\in{\mathbb{N}}^r},\,\,
\bigl(z_{\rho(\alpha)} \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}
\stackrel{d}{=}
\bigl(\tilde{z}_\alpha \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}
\label{Th3eq}$$ and these two arrays are independent of each other.
This result will be more useful to us in a slightly different formulation in terms of the sequence $(V_\alpha)_{\alpha\in {\mathbb{N}}^r}$ in (\[Vs2\]). Namely, if we denote by $$\tilde{V}_\alpha
=
\frac{V_\alpha \exp X_r((z_\beta)_{*\prec \beta\preceq \alpha})}{\sum_{\alpha\in {\mathbb{N}}^r}V_\alpha \exp X_r((z_\beta)_{*\prec \beta\preceq \alpha})}
\label{tildeVs}$$ then the following holds.
\[Th4label\] There exists a random bijection $\rho:{{\cal A}}\to{{\cal A}}$ of the vertices of the tree ${{\cal A}}$, which preserves the parent-child relationship, such that $$\bigl(\tilde{V}_{\rho(\alpha)}\bigr)_{\alpha\in{\mathbb{N}}^r}
\stackrel{d}{=}
\bigl(V_\alpha\bigr)_{\alpha\in{\mathbb{N}}^r},\,\,
\bigl(z_{\rho(\alpha)} \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}
\stackrel{d}{=}
\bigl(\tilde{z}_\alpha \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}
\label{Th4eq}$$ and these two arrays are independent of each other.
**Proof.** We have to apply twice the following simple observation. Suppose that we have a random array $(v_\alpha')_{\alpha\in{\mathbb{N}}^r}$ of positive weights that add up to one and array $(z_\alpha')_{\alpha\in{{\cal A}}\setminus\{*\}}$ generated along the tree similarly to $(\tilde{z}_\alpha)$ above - namely, ${z}_n'$ for $n\in{\mathbb{N}}$ are i.i.d. random variables with some distribution $\nu_0$ and, if we already constructed ${z}_\alpha'$ for $|\alpha|\leq p$ then, given any $\alpha\in {\mathbb{N}}^p$, we generate ${z}_{\alpha n}'$ independently for $n\geq 1$ from some conditional distribution $\nu_p((z_\beta)_{*\prec \beta\preceq \alpha},\, \cdot\,)$, and these are generated independently over different such $\alpha.$ Suppose that $(v_\alpha')_{\alpha\in{\mathbb{N}}^r}$ and $(z_\alpha')_{\alpha\in{{\cal A}}\setminus\{*\}}$ are independent. Consider any random permutation $\rho:{{\cal A}}\to{{\cal A}}$ that preserves the parent-child relationship, which depends only on $(v_\alpha')_{\alpha\in{\mathbb{N}}^r}$, i.e. it is a measurable function of this array. Then the arrays $$\bigl(v_{\rho(\alpha)}' \bigr)_{\alpha\in{\mathbb{N}}^r} \,\,\mbox{ and }\,\, \bigl(z_{\rho(\alpha)}' \bigr)_{\alpha\in{{\cal A}}\setminus\{*\}}$$ are independent and $$\bigl(z_{\rho(\alpha)}' \bigr)_{\alpha\in{{\cal A}}\setminus\{*\}}
\stackrel{d}{=}
\bigl(z_\alpha' \bigr)_{\alpha\in{{\cal A}}\setminus\{*\}}.$$ This is obvious because, conditionally on $\rho$, the array $z_{\rho(\alpha)}'$ is generated exactly like $z_\alpha'$ along the tree, so its conditional distribution does not depend on $\rho$. One example of such permutation $\rho$ is the permutation defined in (\[vsall\]), (\[permute\]), (\[Vs2\]), that sorts the cluster weights indexed by $\alpha\in {{\cal A}}\setminus {\mathbb{N}}^r$ defined by $$v_\alpha' = \sum_{\alpha\prec \beta\in{\mathbb{N}}^r} v_\beta'.
\label{vsallprime}$$ Namely, for each $\alpha\in {{\cal A}}\setminus {\mathbb{N}}^r$, we let $\pi_\alpha: {\mathbb{N}}\to {\mathbb{N}}$ be a bijection such that the sequence $v_{\alpha \pi_\alpha(n)}'$ is decreasing for $n\geq 1$ (we assume that all these cluster weights are different as is the case for the Ruelle probability cascades), let $\pi(*)=*$ and define $$\pi(\alpha n) = \pi(\alpha) \pi_{\pi(\alpha)}(n)
\label{permuteprime}$$ recursively from the root to the leaves of the tree. Let us denote $$\mathrm{Sort}\Bigl(
\bigl(v_\alpha' \bigr)_{\alpha\in{\mathbb{N}}^r}, \bigl(z_\alpha' \bigr)_{\alpha\in{{\cal A}}\setminus\{*\}}
\Bigr)
:=
\Bigl(
\bigl(v_{\pi(\alpha)}' \bigr)_{\alpha\in{\mathbb{N}}^r}, \bigl(z_{\pi(\alpha)}' \bigr)_{\alpha\in{{\cal A}}\setminus\{*\}}
\Bigr).
\label{sort}$$ Notice that this sorting operation depends only on $(v_\alpha')_{\alpha\in{\mathbb{N}}^r}$, so it does not affect the distribution of $(z_\alpha')_{\alpha\in{{\cal A}}\setminus\{*\}}$. Now, let us show how (\[Th3eq\]) implies (\[Th4eq\]). First of all, the permutation $\rho$ in the equation (\[Th4eq\]) is just the sorting operation described above, $$\Bigl(
\bigl(\tilde{V}_{\rho(\alpha)}\bigr)_{\alpha\in{\mathbb{N}}^r},
\bigl(z_{\rho(\alpha)} \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}
\Bigr)
=
\mathrm{Sort}
\Bigl(
\bigl(\tilde{V}_{\alpha}\bigr)_{\alpha\in{\mathbb{N}}^r},
\bigl(z_{\alpha} \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}
\Bigr).$$ Let $\pi$ be the permutation in (\[permute\]), (\[Vs2\]) and, trivially, $$\mathrm{Sort}
\Bigl(
\bigl(\tilde{V}_{\alpha}\bigr)_{\alpha\in{\mathbb{N}}^r},
\bigl(z_{\alpha} \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}
\Bigr)
=
\mathrm{Sort}
\Bigl(
\bigl(\tilde{V}_{\pi^{-1}(\alpha)}\bigr)_{\alpha\in{\mathbb{N}}^r},
\bigl(z_{\pi^{-1}(\alpha)} \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}
\Bigr),$$ since the sorting operation does not depend on how we index the array. On the other hand, by the definition (\[tildeVs\]) and the fact that $V_{\pi^{-1}(\alpha)} = v_\alpha$, $$\tilde{V}_{\pi^{-1}(\alpha)}
=
\frac{v_\alpha \exp X_r((z_{\pi^{-1}(\beta)})_{\beta\in p(\alpha)})}{\sum_{\alpha\in {\mathbb{N}}^r}v_\alpha \exp X_r((z_{\pi^{-1}(\beta)})_{\beta\in p(\alpha)})}.$$ Also, since the permutation $\pi$ depends only on $(v_\alpha)$, by the above observation, the arrays $(v_{\alpha})_{\alpha\in{\mathbb{N}}^r}$ and $(z_{\pi^{-1}(\alpha)})_{\alpha\in{{\cal A}}\setminus\{*\}}$ are independent and $$\bigl(z_{\pi^{-1}(\alpha)} \bigr)_{\alpha\in{{\cal A}}\setminus\{*\}}
\stackrel{d}{=}
\bigl(z_\alpha \bigr)_{\alpha\in{{\cal A}}\setminus\{*\}}.$$ Comparing with the definition (\[tildevs\]), this gives that $$\Bigl(
\bigl(\tilde{V}_{\pi^{-1}(\alpha)}\bigr)_{\alpha\in{\mathbb{N}}^r},
\bigl(z_{\pi^{-1}(\alpha)} \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}
\Bigr)
\stackrel{d}{=}
\Bigl(
\bigl(\tilde{v}_{\alpha}\bigr)_{\alpha\in{\mathbb{N}}^r},
\bigl(z_{\alpha} \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}
\Bigr),$$ and all together we showed that $$\Bigl(
\bigl(\tilde{V}_{\rho(\alpha)}\bigr)_{\alpha\in{\mathbb{N}}^r},
\bigl(z_{\rho(\alpha)} \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}
\Bigr)
\stackrel{d}{=}
\mathrm{Sort}
\Bigl(
\bigl(\tilde{v}_{\alpha}\bigr)_{\alpha\in{\mathbb{N}}^r},
\bigl(z_{\alpha} \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}
\Bigr).$$ Since we already use the notation $\rho$, let us denote the permutation $\rho$ in (\[Th3eq\]) by $\rho'$. Then (\[Th3eq\]) implies $$\begin{aligned}
&
=
\bigl(\tilde{v}_{\rho'(\alpha)}\bigr)_{\alpha\in{\mathbb{N}}^r},
\\
&
\bigl({v}_{\alpha}\bigr)_{\alpha\in{\mathbb{N}}^r},
\bigl(\tilde{z}_{\alpha} \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}
\Bigr)
=
\Bigl(
\bigl({V}_{\alpha}\bigr)_{\alpha\in{\mathbb{N}}^r},
\bigl(\tilde{z}_{\pi(\alpha)} \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}
\Bigr).\end{aligned}$$ Finally, since the sorting permutation $\pi$ depends only on the array $(v_\alpha)$, by the above observation, the array $(\tilde{z}_{\pi(\alpha)})$ is independent of $(V_\alpha)$ and has the same distribution as $(\tilde{z}_{\alpha})$. This finishes the proof.
Cavity equations for the pure states {#Sec3label}
====================================
In this section, we will obtain some general consequences of the cavity equations (\[SC\]) that do not depend on any inductive assumptions. In the next section, we will push this further under the assumption (M) made in Section \[Sec2ilabel\]. First of all, let us rewrite the cavity equations (\[SC\]) taking into account the consequences of the Ghirlanda-Guerra identities in (\[Gdiscrete\]) and (\[sigmaf2\]). Let us define $$\begin{aligned}
{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\alpha_I \ =\ & \ h\bigl( (\omega_\beta)_{\beta\in p(\alpha)}, (\omega_\beta^I)_{\beta\in p(\alpha)} \bigr),
\label{sialpha}
\\
A_i^\alpha({{\varepsilon}}) \ =\ & \sum_{k\leq \pi_i(\lambda K)} \theta_{k,i}( {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\alpha_{1,k,i}, \ldots, {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\alpha_{K-1,k,i}, {{\varepsilon}})
\nonumber
\\
& \ + \theta_i^1({{\varepsilon}})
+\sum_{d\geq 2} \sum_{k\leq \pi_i(d)}\theta_{k,i}^d({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\alpha_{{1,d,k,i}},\ldots, {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\alpha_{{d-1,d,k,i}},{{\varepsilon}}),
\label{Aieps}
\\
A_i^\alpha \ =\ & \log {{\rm Av}}\exp A_i^\alpha({{\varepsilon}}),
\label{Aialpha}
\\
\xi_i^\alpha \ =\ & \frac{{{\rm Av}}{{\varepsilon}}\exp A_i^\alpha({{\varepsilon}}) }{\exp A_i^\alpha},
\label{Sec3xiialpha}\end{aligned}$$ and let $A^\alpha = \sum_{i\leq n} A_i^\alpha$. We will keep the dependence of $A^\alpha$ on $n$ implicit for simplicity of notation. Then (\[Ulbar2\]) can be redefined by (using equality in distribution (\[sigmaf2\])) $${U}_\ell =\, \sum_{\alpha\in {\mathbb{N}}^r} V_\alpha \prod_{i\in C_\ell^1} \xi_i^\alpha \prod_{i\in C_\ell^2} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\alpha\,\exp A^\alpha \ \mbox{ and } \
{V} =\, \sum_{\alpha\in {\mathbb{N}}^r } V_\alpha \exp A^\alpha.
\label{Ulbaralpha}$$ Moreover, if we denote $$\tilde{V}_\alpha = \frac{V_\alpha \exp A^\alpha}{{V}}
= \frac{V_\alpha \exp A^\alpha}{\sum_{\alpha\in {\mathbb{N}}^r} V_\alpha \exp A^\alpha}
\label{Valpha}$$ then the cavity equations (\[SC\]) take form $${\mathbb{E}}\prod_{\ell\leq q} \sum_{\alpha\in {\mathbb{N}}^r} V_\alpha \prod_{i\in C_\ell} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\alpha
={\mathbb{E}}\prod_{\ell\leq q} \sum_{\alpha\in {\mathbb{N}}^r} \tilde{V}_\alpha \prod_{i\in C_\ell^1} \xi_i^\alpha \prod_{i\in C_\ell^2} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\alpha.
\label{SCbaralpha}$$ We can also write this as $${\mathbb{E}}\sum_{\alpha_1,\ldots, \alpha_q} V_{\alpha_1}\cdots V_{\alpha_q} \prod_{\ell\leq q}
\prod_{i\in C_\ell} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}
={\mathbb{E}}\sum_{\alpha_1,\ldots, \alpha_q} \tilde{V}_{\alpha_1} \cdots \tilde{V}_{\alpha_q} \prod_{\ell\leq q} \prod_{i\in C_\ell^1} \xi_i^{\alpha_\ell} \prod_{i\in C_\ell^2} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}.
\label{SCnew}$$ We will now use this form of the cavity equations to obtain a different form directly for the pure states that does not involve averaging over the pure states.
Let us formulate the main result of this section. Let ${{\cal F}}$ be the $\sigma$-algebra generated by the random variables that are not indexed by $\alpha\in {{\cal A}}\setminus \{*\}$, namely, $$\theta_{k,i}, \theta_i^1, \theta^d_{k,i}, \pi_i(\lambda K), \pi_i(d), \omega_*, \omega_*^I
\label{FF}$$ for various indices, excluding the random variables $\omega_{\alpha}$ and $\omega_{\alpha}^I$ that are indexed by $\alpha\in {{\cal A}}\setminus \{*\}$. Let ${{\cal I}}_i$ be the set of indices $I$ that appear in various ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_I^\alpha$ in (\[Aieps\]), i.e. $I$ of the type $(\ell, k,i,)$ or $(\ell, d,k,i)$. Let ${{\cal I}}= \cup_{i\geq 1} {{\cal I}}_i$ and let $$z_\alpha^i = \bigl(\omega_\alpha^I \bigr)_{I\in{{\cal I}}_i},\,\,
z_\alpha = \bigl( \omega_\alpha, (z_\alpha^i)_{i\geq 1} \bigr)= \bigl( \omega_\alpha, (\omega_\alpha^I)_{I\in{{\cal I}}} \bigr),
\label{zeealpha}$$ Notice that with this notation, conditionally on ${{\cal F}}$, the random variables $A_i^\alpha$ and $\xi_i^\alpha$ in (\[Aialpha\]), (\[Sec3xiialpha\]) for $\alpha\in{\mathbb{N}}^r$ can be written as $$\xi_i^\alpha = \xi_i \bigl( (\omega_\beta,z_\beta^i)_{*\prec \beta \preceq \alpha}\bigr),\,\,
A_i^\alpha = \chi_i \bigl( (\omega_\beta, z_\beta^i)_{*\prec \beta \preceq \alpha}\bigr).
\label{xichi}$$ for some function $\xi_i$ and $\chi_i$ (that implicitly depend on the random variables in (\[FF\])) and $$A^\alpha = X\bigl( (z_\beta)_{*\prec \beta \preceq \alpha}\bigr) := \sum_{i\leq n} \chi_i \bigl( (\omega_\beta, z_\beta^i)_{*\prec \beta \preceq \alpha}\bigr).$$ In the setting of the previous section, let $X_r = X$ in (\[Xr\]) and let $(\tilde{z}_\alpha)_{\alpha\in{{\cal A}}\setminus \{*\}}$ be the array generated along the tree using the conditional probabilities (\[ch51transitionprime\]). Recall that this means the following. The definition in (\[ch54Xp\]) can be written as $$X_{|\alpha|-1}\bigl( (z_\beta)_{*\prec \beta \prec \alpha}\bigr)
=\frac{1}{\zeta_{|\alpha|-1}} \log {\mathbb{E}}_{z_\alpha} \exp \zeta_{|\alpha|-1} X_{|\alpha|}\bigl( (z_\beta)_{*\prec \beta \preceq \alpha}\bigr),
\label{Sec3Xp}$$ where ${\mathbb{E}}_{z_\alpha}$ is the expectation in $z_\alpha$, and the definition in (\[ch31Wp\]) can be written as $$W_{|\alpha|-1} \bigl( (z_\beta)_{*\prec \beta \prec \alpha},z_\alpha\bigr)
=
\exp \zeta_{|\alpha|-1}\Bigl(X_{|\alpha|}\bigl( (z_\beta)_{*\prec \beta \preceq \alpha}\bigr) - X_{|\alpha|-1}\bigl( (z_\beta)_{*\prec \beta \prec \alpha}\bigr) \Bigr).
\label{Sec3Walpha}$$ Then the array $(\tilde{z}_\alpha)_{\alpha\in{{\cal A}}\setminus \{*\}}$ is generated along the tree from the root to the leaves according to the conditional probabilities in (\[ch51transitionprime\]), namely, given $(\tilde{z}_\beta)_{*\prec \beta \prec \alpha}$ we generated $\tilde{z}_\alpha$ by the change of density $W_{|\alpha|-1} \bigl( (\tilde{z}_\beta)_{*\prec \beta \prec \alpha},\ \cdot \ \bigr)$. Let us emphasize one more time that this entire construction is done conditionally on ${{\cal F}}$. Also, notice that the coordinates $\omega_\alpha^I$ in (\[zeealpha\]) were independent for different $I$, but the corresponding coordinates $\tilde{\omega}_\alpha^I$ of $
\tilde{z}_\alpha = \bigl( \tilde{\omega}_\alpha, (\tilde{\omega}_\alpha^I)_{I\in{{\cal I}}} \bigr)
$ are no longer independent, because $X_r$ and the changes of density $W_p$ depend on all of them. As in (\[zeealpha\]) and (\[xichi\]), let us denote $$\tilde{z}_\alpha^i = \bigl(\tilde{\omega}_\alpha^I \bigr)_{I\in{{\cal I}}_i},\,\,
\tilde{z}_\alpha = \bigl( \tilde{\omega}_\alpha, (\tilde{z}_\alpha^i)_{i\geq 1} \bigr),\,\,
\tilde{\xi}_i^\alpha = \xi_i \bigl( (\tilde{\omega}_\beta, \tilde{z}_\beta^i)_{*\prec \beta \preceq \alpha}\bigr).
\label{Sec4tildas}$$ We will prove the following.
\[Sec4Th\] The equality in distribution holds (not conditionally on ${{\cal F}}$), $$\bigl({{\tilde{\xi}}}_i^{\alpha} \bigr)_{i\leq n,\alpha\in {\mathbb{N}}^r}
\stackrel{d}{=}
\bigl({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha} \bigr)_{i\leq n,\alpha\in {\mathbb{N}}^r}.
\label{Sec4ThEq}$$
**Proof.** As in (\[Sec2eq1\]) and (\[RabSec2\]), we can write $$R_{\alpha, \beta}= {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
=
=
{\mathbb{E}}_i \, {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\beta},$$ where ${\mathbb{E}}_i$ denotes the expectation in random variables $\omega_\beta^i$ in (\[sialpha\]) that depend on the spin index $i$, and $R_{\alpha,\beta} = q_{\alpha\wedge\beta}$.
In the cavity equations (\[SCnew\]), let us now make a special choice of the sets $C_\ell^2$. For each pair $(\ell,\ell')$ of replica indices such that $1\leq \ell<\ell'\leq q$, take any integer $n_{\ell,\ell'}\geq 0$ and consider a set $C_{\ell,\ell'}\subseteq \{n+1,\ldots,m\}$ of cardinality $|C_{\ell,\ell'}|=n_{\ell,\ell'}$. Let all these sets be disjoint, which can be achieved by taking $m=n+\sum_{1\leq \ell<\ell'\leq q} n_{\ell,\ell'}.$ For each $\ell\leq q$, let $$C_\ell^2 = \Bigl(\bigcup_{\ell'>\ell} C_{\ell,\ell'}\Bigr) \bigcup \Bigl(\bigcup_{\ell'<\ell} C_{\ell',\ell}\Bigr).$$ Then a given spin index $i\in \{n+1,\ldots,m\}$ appears in exactly two sets, say, $C_\ell^2$ and $C_{\ell'}^2$, and the expectation of (\[SCnew\]) in $(\omega_{\beta}^i)$ will produce a factor ${\mathbb{E}}_i \,s_i^{\alpha_\ell} s_i^{\alpha_{\ell'}} = R_{\alpha_\ell,\alpha_{\ell'}}$. For each pair $(\ell,\ell')$, there will be exactly $n_{\ell,\ell'}$ such factors, so averaging in (\[SCnew\]) in the random variables $(\omega_{\beta}^i)$ for all $i\in \{n+1,\ldots,m\}$ will result in $${\mathbb{E}}\sum_{\alpha_1,\ldots, \alpha_q} V_{\alpha_1}\cdots V_{\alpha_q} \prod_{\ell<\ell'} R_{\alpha_\ell, \alpha_{\ell'}}^{n_{\ell,\ell'}}
\prod_{\ell\leq q} \prod_{i\in C_\ell^1} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}
={\mathbb{E}}\sum_{\alpha_1,\ldots, \alpha_q} \tilde{V}_{\alpha_1} \cdots \tilde{V}_{\alpha_q} \prod_{\ell<\ell'} R_{\alpha_\ell, \alpha_{\ell'}}^{n_{\ell,\ell'}} \prod_{\ell\leq q} \prod_{i\in C_\ell^1} \xi_i^{\alpha_\ell}.
\label{SCagain}$$ Approximating by polynomials, we can replace $\prod_{\ell<\ell'} R_{\alpha_\ell, \alpha_{\ell'}}^{n_{\ell,\ell'}}$ by the indicator of the set $${{\cal C}}= \bigl\{(\alpha_1,\ldots, \alpha_q) \ | \ R_{\alpha_\ell, \alpha_{\ell'}} = q_{\ell,\ell'} \mbox{ for all } 1\leq \ell<\ell' \leq q\bigr\}$$ for any choice of constraints $q_{\ell,\ell'}$ taking values in $\{q_0,\ldots,q_r\}$. Therefore, (\[SCagain\]) implies $$\sum_{(\alpha_1,\ldots, \alpha_q)\in {{\cal C}}} {\mathbb{E}}V_{\alpha_1}\cdots V_{\alpha_q} \prod_{\ell\leq q} \prod_{i\in C_\ell^1} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}
= \sum_{(\alpha_1,\ldots, \alpha_q)\in {{\cal C}}} {\mathbb{E}}\tilde{V}_{\alpha_1} \cdots \tilde{V}_{\alpha_q} \prod_{\ell\leq q} \prod_{i\in C_\ell^1} \xi_i^{\alpha_\ell}.
\label{SCF}$$ Using the property (i) above the equation (\[sigmaf2\]), which as we mentioned is the consequence of the Ghirlanda-Guerra identities, we can rewrite the left hand side as $$\sum_{(\alpha_1,\ldots, \alpha_q)\in {{\cal C}}} {\mathbb{E}}V_{\alpha_1}\cdots V_{\alpha_q} \,
{\mathbb{E}}\prod_{\ell\leq q} \prod_{i\in C_\ell^1} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}.$$ Moreover, it is obvious from the definition of the array ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\alpha$ in (\[sialpha\]) that the second expectation depends on $(\alpha_1,\ldots, \alpha_q)\in {{\cal C}}$ only through the overlap constraints $(q_{\ell,\ell'})$, or $(\alpha_\ell\wedge \alpha_{\ell'})$.
On the other hand, on the right hand side of (\[SCF\]) both $\tilde{V}_\alpha$ and $\xi_i^\alpha$ depend on the same random variables through the function $A_i^\alpha({{\varepsilon}})$. If we compare (\[tildeVs\]) and (\[Valpha\]) and apply Theorem \[Th4label\] conditionally on ${{\cal F}}$, we see that that there exists a random bijection $\rho:{{\cal A}}\to{{\cal A}}$ of the vertices of the tree ${{\cal A}}$ which preserves the parent-child relationship and such that $$\bigl(\tilde{V}_{\rho(\alpha)}\bigr)_{\alpha\in{\mathbb{N}}^r}
\stackrel{d}{=}
\bigl(V_\alpha\bigr)_{\alpha\in{\mathbb{N}}^r},\,\,
\bigl( (\xi_i^{\rho(\alpha)} )_{i\leq n} \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}
\stackrel{d}{=}
\bigl( ({{\tilde{\xi}}}_i^{\alpha})_{i\leq n} \bigr)_{\alpha\in {{\cal A}}\setminus \{*\}}$$ and these two arrays are independent of each other (all these statement are conditionally on ${{\cal F}}$). If we denote by ${\mathbb{E}}'$ the conditional expectation given ${{\cal F}}$ then this implies that $$\begin{aligned}
&
\sum_{(\alpha_1,\ldots, \alpha_q)\in {{\cal C}}} {\mathbb{E}}'\, \tilde{V}_{\alpha_1} \cdots \tilde{V}_{\alpha_q} \prod_{\ell\leq q} \prod_{i\in C_\ell^1} \xi_i^{\alpha_\ell}
=
\sum_{(\alpha_1,\ldots, \alpha_q)\in {{\cal C}}} {\mathbb{E}}'\, \tilde{V}_{\rho(\alpha_1)} \cdots \tilde{V}_{\rho(\alpha_q)} \prod_{\ell\leq q} \prod_{i\in C_\ell^1} \xi_i^{\rho(\alpha_\ell)}
\\
&=
\sum_{(\alpha_1,\ldots, \alpha_q)\in {{\cal C}}} {\mathbb{E}}'\, V_{\alpha_1} \cdots V_{\alpha_q} \prod_{\ell\leq q} \prod_{i\in C_\ell^1} {{\tilde{\xi}}}_i^{\alpha_\ell}
=
\sum_{(\alpha_1,\ldots, \alpha_q)\in {{\cal C}}} {\mathbb{E}}\,' V_{\alpha_1} \cdots V_{\alpha_q} \,
{\mathbb{E}}'\, \prod_{\ell\leq q} \prod_{i\in C_\ell^1} {{\tilde{\xi}}}_i^{\alpha_\ell}.\end{aligned}$$ Since the distribution of $(V_\alpha)_{\alpha\in{\mathbb{N}}^r}$ does not depend on the condition and ${\mathbb{E}}\,' V_{\alpha_1} \cdots V_{\alpha_q} = {\mathbb{E}}V_{\alpha_1} \cdots V_{\alpha_q}$, taking the expectation gives $$\sum_{(\alpha_1,\ldots, \alpha_q)\in {{\cal C}}} {\mathbb{E}}\tilde{V}_{\alpha_1} \cdots \tilde{V}_{\alpha_q} \prod_{\ell\leq q} \prod_{i\in C_\ell^1} \xi_i^{\alpha_\ell}
=
\sum_{(\alpha_1,\ldots, \alpha_q)\in {{\cal C}}} {\mathbb{E}}V_{\alpha_1} \cdots V_{\alpha_q} \,
{\mathbb{E}}\prod_{\ell\leq q} \prod_{i\in C_\ell^1} {{\tilde{\xi}}}_i^{\alpha_\ell}.$$ This proves that $$\sum_{(\alpha_1,\ldots, \alpha_q)\in {{\cal C}}} {\mathbb{E}}V_{\alpha_1}\cdots V_{\alpha_q} \,
{\mathbb{E}}\prod_{\ell\leq q} \prod_{i\in C_\ell^1} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}
=
\sum_{(\alpha_1,\ldots, \alpha_q)\in {{\cal C}}} {\mathbb{E}}V_{\alpha_1} \cdots V_{\alpha_q} \,
{\mathbb{E}}\prod_{\ell\leq q} \prod_{i\in C_\ell^1} {{\tilde{\xi}}}_i^{\alpha_\ell}.$$ Again, the second expectation in the sum on the right depends on $(\alpha_1,\ldots, \alpha_q)\in {{\cal C}}$ only through the overlap constraints $(q_{\ell,\ell'})$ and, since the choice of the constraints was arbitrary, we get $${\mathbb{E}}\prod_{\ell\leq q} \prod_{i\in C_\ell^1} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}
=
{\mathbb{E}}\prod_{\ell\leq q} \prod_{i\in C_\ell^1} {{\tilde{\xi}}}_i^{\alpha_\ell}$$ for any $\alpha_1,\ldots, \alpha_q \in {\mathbb{N}}^r$. Clearly, one can express any joint moment of the elements in these two arrays by choosing $q\geq 1$ large enough and choosing $\alpha_1,\ldots, \alpha_q$ and the sets $C_\ell^1$ properly, so the proof is complete.
A consequence of the cavity equations for the pure states {#Sec4label}
=========================================================
We will continue using the notation of the previous section, only in this section we will take $n=2$ in Theorem \[Sec4Th\]. Let us recall the assumption (M) made at the end of Section \[Sec2ilabel\]: we consider some $[p+1]\preceq \alpha_1,\ldots,\alpha_k \in {\mathbb{N}}^r$ such that the following holds:
1. For any subset $S\subseteq \{1,\ldots, k\}$, ${{\mathbb{E}_{[p],i}}}\prod_{\ell\in S}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}$ does not depend on $\omega_{[p+1]}$.
In this section, we will obtain a further consequence of the cavity equations using that ${{\mathbb{E}_{[p],i}}}\prod_{\ell\leq k}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}$ does not depend on $\omega_{[p+1]}$, but a similar consequence will hold for any subset of these vertices.
Let us denote by ${{\mathbb{E}_{[p]}}}$ the expectation with respect to the random variables $\omega_\eta, \omega_\eta^I$ indexed by the ancestors $\eta\succ [p]$ of $[p]$. We will use the same notation ${{\mathbb{E}_{[p]}}}$ to denote the expectation with respect to the random variables $\tilde{\omega}_\eta, \tilde{\omega}_\eta^I$ for $\eta\succ [p]$ conditionally on $\tilde{\omega}_\eta, \tilde{\omega}_\eta^I$ for $\eta\preceq [p]$ and all other random variables that generate the $\sigma$-algebra ${{\cal F}}$ in (\[FF\]). Given any finite set ${C}\subset {\mathbb{N}}^r$, let us denote $${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{C} = \prod_{\alpha\in {C}} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\alpha_i,\,\,
{{\tilde{\xi}}}_i^{C} = \prod_{\alpha\in {C}} {{\tilde{\xi}}}^\alpha_i
\,\,\mbox{ and }\,\,
\xi_i^{C} = \prod_{\alpha\in {C}} \xi^\alpha_i.$$ Then the following holds for $[p+1]\preceq \alpha_1,\ldots,\alpha_k \in {\mathbb{N}}^r$.
\[Sec5Lem1\] If ${C} = \{\alpha_1,\ldots,\alpha_k\}$ and ${{\mathbb{E}_{[p],i}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{C}$ does not depend on $\omega_{[p+1]}$ then $${{\mathbb{E}_{[p]}}}{{\tilde{\xi}}}_1^{{C}} {{\tilde{\xi}}}_2^{{C}} = {{\mathbb{E}_{[p]}}}{{\tilde{\xi}}}_1^{{C}} {{\mathbb{E}_{[p]}}}{{\tilde{\xi}}}_2^{{C}}
\label{Sec3Lem1eq}$$ almost surely.
**Proof.** First of all, $${{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_1^{{C}} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}} =
{{\mathbb{E}_{[p]}}}\prod_{i\leq 2 } {{\mathbb{E}_{[p],i}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{{C}}
= \prod_{i\leq 2 } {{\mathbb{E}_{[p],i}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{{C}}$$ almost surely, since ${{\mathbb{E}_{[p],i}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{C}$ does not depend on $\omega_{[p+1]}$. Similarly, $${{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{{C}} =
{{\mathbb{E}_{[p]}}}{{\mathbb{E}_{[p],i}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{{C}}
= {{\mathbb{E}_{[p],i}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{{C}}$$ almost surely and, therefore, $$\begin{aligned}
0
= &\
{\mathbb{E}}\bigl({{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_1^{{C}} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}} - {{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_1^{{C}} {{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}} \bigr)^2
\nonumber
\\
= &\
{\mathbb{E}}\bigl({{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_1^{{C}} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}}\bigr)^2
-2 {\mathbb{E}}\bigl({{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_1^{{C}} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}}\bigr) \bigl({{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_1^{{C}} {{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}}\bigr)
+{\mathbb{E}}\bigl({{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_1^{{C}} {{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}}\bigr)^2.
\label{Sec3above}\end{aligned}$$ Let us now rewrite each of these terms using replicas. Let ${C}_1 = {C}$ and for $j=2,3,4$ let ${C}_j = \{\alpha^j_1,\ldots,\alpha^j_k\}$ for arbitrary $[p+j]\preceq \alpha^j_1,\ldots,\alpha^j_k \in {\mathbb{N}}^r$ such that $\alpha^j_\ell \wedge \alpha^j_{\ell'} = \alpha_\ell \wedge \alpha_{\ell'}$ for any $\ell,\ell'\leq k.$ In other words, ${C}_j$ are copies of ${C}$ that consists of the descendants of different children of $[p]$. Therefore, we can write $${{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_1^{{C}_j} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}_j}
=
{{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_1^{{C}_{j'}} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}_{j'}}
\,\,\mbox{ and }\,\,
{{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{{C}_j}
=
{{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{{C}_{j'}}$$ almost surely for any $j,j'\leq 4$ and $$\begin{aligned}
{\mathbb{E}}\bigl({{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_1^{{C}} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}}\bigr)^2
= &\
{\mathbb{E}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_1^{{C}_1} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}_1}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_1^{{C}_2} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}_2},
\\
{\mathbb{E}}\bigl({{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
= &\
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}_3},
\\
{\mathbb{E}}\bigl({{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_1^{{C}} {{\mathbb{E}_{[p]}}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}}\bigr)^2
= &\
{\mathbb{E}}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_1^{{C}_1} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}_2}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_1^{{C}_3} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_2^{{C}_4}.\end{aligned}$$ By Theorem \[Sec4Th\], this and (\[Sec3above\]) imply that $${\mathbb{E}}{{\tilde{\xi}}}_1^{{C}_1} {{\tilde{\xi}}}_2^{{C}_1}{{\tilde{\xi}}}_1^{{C}_2} {{\tilde{\xi}}}_2^{{C}_2}
-2 {\mathbb{E}}{{\tilde{\xi}}}_1^{{C}_1} {{\tilde{\xi}}}_2^{{C}_1}{{\tilde{\xi}}}_1^{{C}_2} {{\tilde{\xi}}}_2^{{C}_3}
+{\mathbb{E}}{{\tilde{\xi}}}_1^{{C}_1} {{\tilde{\xi}}}_2^{{C}_2}{{\tilde{\xi}}}_1^{{C}_3} {{\tilde{\xi}}}_2^{{C}_4} = 0.$$ Repeating the above computation backwards for ${{\tilde{\xi}}}$ instead of ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}$ gives $${\mathbb{E}}\bigl({{\mathbb{E}_{[p]}}}{{\tilde{\xi}}}_1^{{C}} {{\tilde{\xi}}}_2^{{C}} - {{\mathbb{E}_{[p]}}}{{\tilde{\xi}}}_1^{{C}} {{\mathbb{E}_{[p]}}}{{\tilde{\xi}}}_2^{{C}} \bigr)^2 = 0$$ and this finishes the proof.
By analogy with (\[Sec2Walpha\]) and (\[Sec2expectW\]), let us rewrite the expectation ${{\mathbb{E}_{[p]}}}$ with respect to the random variables $\tilde{\omega}_\alpha, \tilde{\omega}_\alpha^I$ for $\alpha\succ [p]$ in terms of the expectation with respect to the random variables ${\omega}_\alpha, {\omega}_\alpha^I$ for $\alpha\succ [p]$, writing explicitly the changes of density $$W_\alpha = W_{|\alpha|-1} \bigl( (z_\beta)_{*\prec \beta \prec \alpha},z_\alpha\bigr).
\label{Sec5Wa}$$ As in Lemma \[Sec5Lem1\], let ${C} = \{\alpha_1,\ldots,\alpha_k\}$ for some $[p+1]\preceq \alpha_1,\ldots,\alpha_k \in {\mathbb{N}}^r$, let $$p([p],{C}) = \bigl\{\beta \ \bigr|\ [p+1]\preceq \beta\preceq \alpha, \alpha\in{C} \bigr\}$$ and define $$W_{[p],{C}} = \prod_{\alpha\in p([p],{C})} W_\alpha.
\label{Sec5WpC}$$ With this notation, we can rewrite (\[Sec3Lem1eq\]) as $${{\mathbb{E}_{[p]}}}\xi_1^{{C}} \xi_2^{{C}} W_{[p],{C}} = {{\mathbb{E}_{[p]}}}\xi_1^{{C}}W_{[p],{C}}\, {{\mathbb{E}_{[p]}}}\xi_2^{{C}}W_{[p],{C}}
\label{Sec5eq1}$$ almost surely. Notice that in (\[Sec3Lem1eq\]) almost surely meant for almost all random variables in (\[FF\]) that generate the $\sigma$-algebra ${{\cal F}}$ and for almost all $\tilde{z}_\alpha$ for $\alpha\preceq [p]$ that are generated conditionally on ${{\cal F}}$ according to the changes of density in (\[Sec3Walpha\]). However, even though in (\[Sec5eq1\]) we simply expressed the expectation with respect to $\tilde{z}_\alpha$ for $[p]\prec \alpha$ using the changes of density explicitly, after this averaging both sides depend on ${z}_\alpha$ for $\alpha\preceq [p]$, so almost surely now means for almost all random variables in (\[FF\]) that generate the $\sigma$-algebra ${{\cal F}}$ and for almost all ${z}_\alpha$ for $\alpha\preceq [p]$. The reason we can do this is very simple. Notice that $A_i^\alpha({{\varepsilon}})$ in (\[Aieps\]) can be bounded by $$|A_i^\alpha({{\varepsilon}})| \leq c_i:= \sum_{k\leq \pi_i(\lambda K)} \|\theta_{k,i}\|_{\infty} + |g_i^1| +\sum_{d\geq 2} \sum_{k\leq \pi_i(d)}|g_{k,i}^d|,$$ which, by the assumption (\[gsvar\]), is almost surely finite (notice also that $c_i$ are ${{\cal F}}$-measurable). By induction in (\[Sec3Xp\]), all $|X_{|\alpha|}|\leq c=c_1+c_2$ almost surely and, therefore, all changes of density in (\[Sec3Walpha\]) satisfy $e^{-2c}\leq W_{|\alpha|}\leq e^{2c}$ almost surely. Therefore, conditionally on ${{\cal F}}$, the distribution of all $z_\alpha$ and $\tilde{z}_\alpha$ are absolutely continuous with respect to each other and, therefore, we can write almost surely equality in (\[Sec5eq1\]) in terms of the random variables $z_\alpha$ for $\alpha\preceq [p]$.
Next, we will reformulate (\[Sec5eq1\]) using the assumption (\[indass\]). To simplify the notation, let us denote for any $\alpha\in {{\cal A}}\setminus\{*\}$, $$\omega_{\preceq\alpha} = (\omega_\beta)_{*\prec \beta\preceq \alpha},\,\,
z_{\preceq\alpha}^i = (z_\beta^i)_{*\prec \beta \preceq \alpha},\,\,
z_{\preceq\alpha} = (z_\beta)_{*\prec \beta \preceq \alpha}$$ and define $\omega_{\prec\alpha}, z_{\prec\alpha}^i $ and $z_{\prec\alpha}$ similarly. Then, we can rewrite (\[xichi\]) for $[p+1]\preceq \alpha\in {\mathbb{N}}^r$ as $$\xi_i^{\alpha} = \xi_i \bigl( \omega_{\preceq [p+1]}, z^i_{\preceq \alpha}\bigr),\,\,
A_i^{\alpha} =\chi_i \bigl( \omega_{\preceq [p+1]}, z^i_{\preceq \alpha}\bigr).
\label{Sec5xichi}$$ Since in the previous section we set $n=2$, we have $$A^{\alpha} = \sum_{i\leq 2} \chi_i \bigl( \omega_{\preceq [p+1]}, z^i_{\preceq \alpha}\bigr).$$ Because of the absence of the random variables $\omega_\alpha$ for $[p+1]\prec \alpha$, the integration in $z_\alpha$ in the recursive definition (\[Sec3Xp\]) will decouple when $[p+1]\prec \alpha$ into integration over $z_\alpha^1$ and $z_\alpha^2$. Namely, let $\chi_{i,r}=\chi_i$ and, for $[p+1] \preceq \alpha$, let us define by decreasing induction on $|\alpha|$, $$\chi_{i,|\alpha|-1}\bigl( \omega_{\preceq [p+1]}, z^i_{\prec \alpha }\bigr)
=
\frac{1}{\zeta_{|\alpha|-1}} \log {\mathbb{E}}_{z_{\alpha}^i} \exp \zeta_{|\alpha|-1} \chi_{i,|\alpha|}\bigl( \omega_{\preceq [p+1]}, z^i_{\preceq \alpha}\bigr).
\label{chij}$$ First of all, for $[p+1]\prec \alpha$, by decreasing induction on $|\alpha|$, $$\begin{aligned}
X_{|\alpha|-1}\bigl( z_{\prec \alpha}\bigr)
= &\
\frac{1}{\zeta_{|\alpha|-1}} \log {\mathbb{E}}_{z_{\alpha}} \exp \zeta_{|\alpha|-1} X_{|\alpha|}\bigl( z_{\preceq \alpha}\bigr)
\nonumber
\\
\{\mbox{induction assumption}\}= &\
\frac{1}{\zeta_{|\alpha|-1}} \log {\mathbb{E}}_{z_{\alpha}} \exp \zeta_{|\alpha|-1}
\sum_{i\leq 2}
\chi_{i,|\alpha|}\bigl( \omega_{\preceq [p+1]}, z^i_{\preceq \alpha}\bigr)
\nonumber
\\
\{\mbox{independence}\}= &\
\frac{1}{\zeta_{|\alpha|-1}} \log \prod_{i\leq 2} {\mathbb{E}}_{z_{\alpha}^i} \exp \zeta_{|\alpha|-1}
\chi_{i,|\alpha|}\bigl( \omega_{\preceq [p+1]}, z^i_{\preceq \alpha}\bigr)
\nonumber
\\
\{\mbox{definition (\ref{chij})}\}= &\
\sum_{i\leq 2}
\chi_{i,|\alpha|-1}\bigl( \omega_{\preceq [p+1]}, z^i_{\prec \alpha}\bigr).
\label{Xichii}\end{aligned}$$ When we do the same computation for $\alpha = [p+1]$, the expectation ${\mathbb{E}}_{z_{[p+1]}}$ also involves $\omega_{[p+1]}$, so we end up with $$\begin{aligned}
X_{p}\bigl( z_{\preceq [p]}\bigr)
= &\
\frac{1}{\zeta_p} \log {\mathbb{E}}_{\omega_{[p+1]}}\prod_{i\leq 2} {\mathbb{E}}_{z_{[p+1]}^i} \exp \zeta_p
\chi_{i,p+1}\bigl( \omega_{\preceq [p+1]}, z^i_{\preceq [p+1]}\bigr)
\nonumber
\\
\{\mbox{definition (\ref{chij})}\}
= &\
\frac{1}{\zeta_p} \log {\mathbb{E}}_{\omega_{[p+1]}} \exp \zeta_p
\sum_{i\leq 2}
\chi_{i,p}\bigl( \omega_{\preceq [p+1]}, z^i_{\preceq [p]}\bigr).
\label{Sec5Xp}\end{aligned}$$ For $[p+1]\preceq \alpha$, let us define for $i=1,2$, $$W_{|\alpha|-1}^i \bigl( \omega_{\preceq [p+1]}, z^i_{\preceq \alpha}\bigr)
=
\exp \zeta_{|\alpha|-1}\Bigl(\chi_{i,|\alpha|}\bigl( \omega_{\preceq [p+1]}, z^i_{\preceq \alpha}\bigr) - \chi_{i,|\alpha|-1}\bigl( \omega_{\preceq [p+1]}, z^i_{\prec \alpha}\bigr) \Bigr).
\label{Sec5Walphai}$$ Comparing this with the definition (\[ch31Wp\]) and using (\[Xichii\]) we get that for $[p+1]\prec \alpha$, $$W_{|\alpha|-1} \bigl( z_{\preceq \alpha} \bigr)
=
\prod_{i\leq 2}
W_{|\alpha|-1}^i \bigl( \omega_{\preceq [p+1]}, z^i_{\preceq [j+1]}\bigr).
\label{Sec5Walpha}$$ For $\alpha = [p+1]$ this is no longer true, but if we denote $${{{ \sbox{\myboxA}{$\m@thW$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{-0.6mm}}_{p}}= {{{ \sbox{\myboxA}{$\m@thW$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{-0.6mm}}_{p}}\bigl( \omega_{ [p+1]}, z_{\preceq [p]}\bigr) =
\exp \zeta_p\Bigl(\sum_{i\leq 2}\chi_{i,p}\bigl( \omega_{\preceq [p+1]}, z^i_{\preceq [p]}\bigr) - X_{p}(z_{\preceq [p]}) \Bigr)
\label{Sec5Q}$$ then we can write $$W_p \bigl( z_{\preceq [p+1]} \bigr)
=
{{{ \sbox{\myboxA}{$\m@thW$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{-0.6mm}}_{p}}\bigl( \omega_{ [p+1]}, z_{\preceq [p]}\bigr)
\prod_{i\leq 2}
W_p^i \bigl( \omega_{\preceq [p+1]}, z^i_{\preceq [p+1]}\bigr).
\label{Sec5Wp}$$ If, similarly to (\[Sec5Wa\]) and (\[Sec5WpC\]), we denote $$W_\alpha^i = W_{|\alpha|-1}^i \bigl( z_{\preceq \alpha}\bigr)
\,\,\mbox{ and }\,\,
W_{[p],{C}}^i = \prod_{\alpha\in p([p],{C})} W_\alpha^i
\label{Sec5WpCi}$$ then we can rewrite (\[Sec5WpC\]) as $$W_{[p],{C}} = {{{ \sbox{\myboxA}{$\m@thW$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{-0.6mm}}_{p}}\prod_{i\leq 2} W_{[p],{C}}^{i}.
\label{Sec5Wpr}$$ Notice that ${{{ \sbox{\myboxA}{$\m@thW$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{-0.6mm}}_{p}}$ does not depend on $z_\beta^i$ for $[p]\prec \beta$, while $W_{[p],{C}}^{1}$ and $\xi_1^{{C}}$ in (\[Sec5eq1\]) do not depend on $z_\beta^2$ for $[p]\prec \beta$ and $W_{[p],{C}}^{2}$ and $\xi_2^{{C}}$ do not depend on $z_\beta^1$ for $[p]\prec \beta$. This means that if we denote by ${{\mathbb{E}_{[p],i}}}$ the expectation in the random variables $z_\beta^i$ for $[p]\prec \beta$ and denote $$\eta_i^{{C}} = {{\mathbb{E}_{[p],i}}}\, \xi_i^{{C}} W_{[p],{C}}^{i}
\label{Sec5eqsecond}$$ then (\[Sec5eq1\]) can be rewritten as $${\mathbb{E}}_{\omega_{[p+1]}} \eta_1^{{C}} \eta_2^{{C}} {{{ \sbox{\myboxA}{$\m@thW$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{-0.6mm}}_{p}}= {\mathbb{E}}_{\omega_{[p+1]}} \eta_1^{{C}} {{{ \sbox{\myboxA}{$\m@thW$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{-0.6mm}}_{p}}\, {\mathbb{E}}_{\omega_{[p+1]}} \eta_2^{{C}} {{{ \sbox{\myboxA}{$\m@thW$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{-0.6mm}}_{p}}\label{Sec5eq2}$$ almost surely, because after averaging ${{\mathbb{E}_{[p],i}}}$ in $z_\beta^i$ for $[p]\prec \beta$, the only random variable left to be averaged in ${{\mathbb{E}_{[p]}}}$ is $\omega_{[p+1]}$. So far we have just rewritten the equation (\[Sec5eq1\]) under the induction assumption (\[indass\]). Now, we will use this to prove the main result of this section.
Let us make the following simple observation: recalling the definition of $A_i^\alpha({{\varepsilon}})$ in (\[Aieps\]), if we set all but finite number of random variables $(\pi_i(d))_{d\geq 2}$ to zero, the equation (\[Sec5eq2\]) still holds almost surely. To see this, first of all, notice that because the random variables $\pi_i(d)$ take any natural value with positive probability, we can set a finite number of them to any values we like in (\[Sec5eq2\]). For example, for any $D,D'\geq 2$, we can set $\pi_1(d)=\pi_2(d)=n_d$ for $d\leq D$ and set $\pi_1(d)=\pi_2(d)=0$ for $D< d< D'$. The remaining part of the last term in $A_i^\alpha({{\varepsilon}})$ can be bounded uniformly by $$\sum_{d\geq D'} \sum_{k\leq \pi_i(d)} \| \theta_{k,i}^d \|_\infty
\leq
\sum_{d\geq D'} \sum_{k\leq \pi_i(d)}|g_{k,i}^d|,$$ where, by the assumption (\[gsvar\]), we have ${\mathbb{E}}(g_{k,i}^d)^2 \leq 2^{-d}\epsilon^{\mathrm{pert}},$ which implies that this sum goes to zero almost surely as $D'$ goes to infinity. It follows immediately from this that we can set all but finite number of $\pi_i(d)$ in (\[Sec5eq2\]) to zero. Moreover, we will set $\pi_1(\lambda K) = \pi_2(\lambda K) = 0$, since the terms coming from the model Hamiltonian will play no role in the proof - all the information we need is encoded into the perturbation Hamiltonian. From now on we will assume that in (\[Sec5eq2\]), for a given $D\geq 2$, $$\pi_1(\lambda K) = \pi_2(\lambda K) = 0,\,\,
\pi_1(d)=\pi_2(d)=n_d \,\,\mbox{ for }\,\, d\leq D,\,\,
\pi_1(d)=\pi_2(d)=0 \,\,\mbox{ for }\,\, d>D.
\label{fixPoisson}$$ In addition, let us notice that both sides of (\[Sec5eq2\]) are continuous functions of the variables $g_i^1$ and $g_{k,i}^d$ for $k\leq n_d$ for $d\leq D$, $i=1,2$. This implies that almost surely over other random variables the equation (\[Sec5eq2\]) holds for all $g_{k,i}^d$ and, in particular, we can set them to be equal to any prescribed values, $$g_1^1 = g_2^1 = g^1,\,\,
g_{k,1}^d=g_{k,2}^d=g_k^d.
\label{fixgs}$$ The following is the main result of this section.
\[Sec5Th\] The random variables $\eta_i^{{C}}$ do not depend on $\omega_{[p+1]}$.
Here and below, when we say that a function (or random variable) does not depend on a certain coordinate, this means that the function is equal to the average over that coordinate almost surely. In this case, we want to show that $$\eta_i^{{C}} = {\mathbb{E}}_{\omega_{[p+1]}} \eta_i^{{C}}$$ almost surely.
**Proof of Theorem \[Sec5Th\].** Besides the Poisson and Gaussian random variables in (\[fixPoisson\]), (\[fixgs\]) and the random variable $\omega_{[p+1]}$ over which we average in (\[Sec5eq2\]), the random variables $\eta_i^{{C}}$ for $i=1,2$ depend on $\omega_*$, $\omega_{\preceq [p]}$, $(\omega_*^I)_{I\in {{\cal I}}_i}$ and $z_{\preceq [p]}^i$, and ${{{ \sbox{\myboxA}{$\m@thW$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{-0.6mm}}_{p}}$ depends on the same random variables for both $i=1,2$. Let us denote $$u_i = \bigl( (\omega_*^I)_{I\in {{\cal I}}_i}, z_{\preceq [p]}^i \bigr).$$ We already stated that for almost all $\omega_*$, $\omega_{\preceq [p]}$, $u_1$ and $u_2$, the equation (\[Sec5eq2\]) holds for all Poisson and Gaussian random variables fixed as in (\[fixPoisson\]), (\[fixgs\]). Therefore, for almost all $\omega_*$, $\omega_{\preceq [p]}$ the equation (\[Sec5eq2\]) holds for almost all $u_1$, $u_2$ and for all Poisson and Gaussian random variables fixed as in (\[fixPoisson\]), (\[fixgs\]). Let us fix any such $\omega_*$, $\omega_{\preceq [p]}$. Then, we can write $$\eta_i^{{C}} = \varphi(u_i,\omega_{[p+1]}) \ \mbox{ and}\
{{{ \sbox{\myboxA}{$\m@thW$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{-0.6mm}}_{p}}= \psi(u_1,u_2,\omega_{[p+1]})$$ for some functions $\varphi$ and $\psi$. These functions depend implicitly on all the random variables we fixed, and the function $\varphi$ is the same for both $\eta_1^{{C}}$ and $\eta_2^{{C}}$ because we fixed all Poisson and Gaussian random variables in (\[fixPoisson\]), (\[fixgs\]) to be the same for $i=1,2$. The equation (\[Sec5eq2\]) can be written as (in the rest of this proof, let us for simplicity of notation write $\omega$ instead of $\omega_{[p+1]}$) $${\mathbb{E}}_{\omega} \varphi(u_1,\omega) \varphi(u_2,\omega) \psi(u_1,u_2,\omega)
=
\prod_{i=1,2}{\mathbb{E}}_{\omega} \varphi(u_i,\omega) \psi(u_1,u_2,\omega)
\label{Sec5eq3}$$ for almost all $u_1,u_2$. We want to show that $\varphi(u,\omega)$ does not depend on $\omega$. If we denote $$c =: |g_1^1| + |g_2^1| + \sum_{d\leq D} \sum_{k\leq n_d} |g_{k}^d|$$ then, by (\[fixPoisson\]) and (\[fixgs\]), we can bound $A_i^\alpha({{\varepsilon}})$ in (\[Aieps\]) by $|A_i^\alpha({{\varepsilon}})| \leq c$ for $i=1,2$. By induction in (\[chij\]), $|\chi_{i,|\alpha|}|\leq c$ and, by (\[Sec5Xp\]), $|X_{p}|\leq 2c$. Therefore, from the definition of ${{{ \sbox{\myboxA}{$\m@thW$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{-0.6mm}}_{p}}$ in (\[Sec5Q\]), $$e^{-4c}\leq {{{ \sbox{\myboxA}{$\m@thW$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{-0.6mm}}_{p}}= \psi(u_1,u_2,\omega) \leq e^{4c}.
\label{Sec5eq5}$$ Of course, $|\varphi|\leq 1$. Suppose that for some ${{\varepsilon}}>0$, there exists a set $U$ of positive measure such that the variance $\mbox{Var}_{\omega}(\varphi(u,\omega))\geq {{\varepsilon}}$ for $u\in U.$ Given $\delta>0$, let $(S_\ell)_{\ell\geq 1}$ be a partition of $L^1([0,1],d\omega)$ such that $\mbox{diam}(S_\ell)\leq \delta$ for all $\ell.$ Let $$U_\ell = \bigl\{ u \ |\ \varphi(u,\,\cdot\,) \in S_\ell \bigr\}.$$ For some $\ell$, the measure of $U\cap U_\ell$ will be positive, so for some $u_1,u_2\in U$, $${\mathbb{E}}_\omega | \varphi(u_1,\omega) - \varphi(u_2,\omega) | \leq \delta.
\label{Sec5eq6}$$ The equations (\[Sec5eq5\]) and (\[Sec5eq6\]) imply that $$\bigl|
{\mathbb{E}}_\omega \varphi(u_1,\omega) \psi(u_1,u_2,\omega)
- {\mathbb{E}}_\omega \varphi(u_2,\omega) \psi(u_1,u_2,\omega)
\bigr|
\leq e^{4c}\delta$$ and, similarly, $$\bigl|
{\mathbb{E}}_\omega \varphi(u_1,\omega) \varphi(u_2,\omega) \psi(u_1,u_2,\omega)
- {\mathbb{E}}_\omega \varphi(u_1,\omega)^2 \psi(u_1,u_2,\omega)
\bigr|
\leq e^{4c}\delta.$$ Since $|\varphi|\leq 1$ and ${\mathbb{E}}_\omega \psi =1$, the first inequality implies that $$\Bigl|
\prod_{i=1,2}{\mathbb{E}}_\omega \varphi(u_i,\omega) \psi(u_1,u_2,\omega)
- \bigl({\mathbb{E}}_\omega \varphi(u_1,\omega) \psi(u_1,u_2,\omega) \bigr)^2
\Bigr|
\leq
e^{4c}\delta,$$ which, together with the second inequality and (\[Sec5eq3\]), implies $${\mathbb{E}}_\omega \varphi(u_1,\omega)^2 \psi(u_1,u_2,\omega)
-
\bigl({\mathbb{E}}_\omega \varphi(u_1,\omega) \psi(u_1,u_2,\omega) \bigr)^2
\leq 2 e^{4c}\delta.$$ The left hand side is a variance with the density $\psi$ and can be written using replicas as $$\frac{1}{2} \iint\! \bigl(\varphi(u_1,x)- \varphi(u_1,y)\bigr)^2 \psi(u_1,u_2,x)\psi(u_1,u_2,y)\,dx dy.$$ By (\[Sec5eq5\]) and the fact that $u_1\in U$, we can bound this from below by $$\frac{1}{2}e^{-8c} \iint\! \bigl(\varphi(u_1,x)- \varphi(u_1,y)\bigr)^2 \,dx dy
=
e^{-8c}\mbox{Var}_{\omega}(\varphi(u_1,\omega)) \geq e^{-8c}{{\varepsilon}}.$$ Comparing lower and upper bounds, $e^{-8c}{{\varepsilon}}\leq e^{4c}\delta$, we arrive at contradiction, since $\delta>0$ was arbitrary. Therefore, $\mbox{Var}_{\omega}(\varphi(u,\omega)) = 0$ for almost all $u$ and this finishes the proof.
A representation formula via properties of the RPC {#Sec6label}
==================================================
Let us summarize what we proved in the previous section. We considered ${C} = \{\alpha_1,\ldots,\alpha_k\}$ for some $[p+1]\preceq \alpha_1,\ldots,\alpha_k \in {\mathbb{N}}^r$ and assumed that ${{\mathbb{E}_{[p],i}}}\prod_{\ell\leq k}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}$ does not depend on $\omega_{[p+1]}$. Then, as a consequence of this and the cavity equations, we showed that $$\eta_i^{{C}} = {{\mathbb{E}_{[p],i}}}\, \xi_i^{{C}} W_{[p],{C}}^{i}
\label{Sec6eqsecond}$$ also does not depend on $\omega_{[p+1]}$ almost surely, where $$W_{[p],{C}}^i = \prod_{\alpha\in p([p],{C})} W_\alpha^i
\label{Sec6WpCi}$$ and $p([p],{C}) = \bigl\{\beta \ \bigr|\ [p+1]\preceq \beta\preceq \alpha, \alpha\in{C} \bigr\}$. Moreover, this holds for Poisson and Gaussian random variables fixed to arbitrary values as in (\[fixPoisson\]), (\[fixgs\]). By the assumption (M), the same statement holds in we replace ${C}$ by any subset ${C'}\subseteq C$.
In this section, we will represent the expectation ${{\mathbb{E}_{[p],i}}}$ in (\[Sec6eqsecond\]) with respect to $z_\alpha^i$ for $[p]\prec \alpha$ by using the property of the Ruelle probability cascades in Theorem \[Th4label\]. Essentially, the expectation in (\[Sec6eqsecond\]) is of the same type as (\[Sec2expectW\]) if we think of the vertex $[p]$ as a root. Indeed, we are averaging over random variables indexed by the vertices $[p]\prec \alpha$ which form a tree (if we include the root $[p]$) isomorphic to a tree $
\Gamma = {\mathbb{N}}^0\cup {\mathbb{N}}^1 \cup \ldots \cup {\mathbb{N}}^{r-p}
$ of depth $r-p$. We can identify a vertex $[p]\preceq \alpha \in {{\cal A}}$ with the vertex $\{*\} \preceq \gamma \in \Gamma$ such that $\alpha = [p]\gamma$ (for simplicity, we denote by $[p]\gamma$ the concatenation $([p],\gamma)$). Similarly to (\[indass\]), let us define for $\gamma\in{\mathbb{N}}^{r-p}$, $${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_I^\gamma = h\bigl( (\omega_\beta)_{\beta\preceq [p]},\omega_{[p+1]}, (\omega_\beta^I)_{\beta\preceq [p]},(\omega_{[p]\beta}^I)_{*\prec \beta \preceq \gamma} \bigr).
\label{Sec6sialpha}$$ Notice a subtle point here: the random variables ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_I$ are not exactly the same as ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\alpha_I$ in (\[indass\]) for $\alpha = [p]\gamma$. They are exactly the same only if $[p+1]\preceq \alpha$, and in this case we will often write ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_I^\gamma= {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_I^{[p]\gamma}.$ Otherwise, if $[p+j]\preceq \alpha$ for $j\geq 2$ then in (\[indass\]) we plug in the random variable $\omega_{[p+j]}$ instead of $\omega_{[p+1]}$ as we did in (\[Sec6sialpha\]). The reason for this will become clear soon but, basically, we are going to represent the average ${{\mathbb{E}_{[p],i}}}$ in (\[Sec6eqsecond\]) with respect to $\omega_{[p]\beta}^I$ for $\{*\}\prec \beta$ using the Ruelle probability cascades while $\omega_{[p+1]}$ appears in (\[Sec6eqsecond\]) on the outside of this average.
Similarly to (\[Aieps\]) – (\[Sec3xiialpha\]), let us define for $\gamma\in {\mathbb{N}}^{r-p}$, $$\begin{aligned}
A_i^\gamma({{\varepsilon}}) \ =\ & \theta^1({{\varepsilon}})
+\sum_{2\leq d\leq D} \sum_{k\leq n_d}\theta_{k}^d({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{{1,d,k,i}},\ldots, {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{{d-1,d,k,i}},{{\varepsilon}}),
\label{Sec6Aieps}
\\
A_i^\gamma \ =\ & \log {{\rm Av}}\exp A_i^\gamma({{\varepsilon}}),
\label{Sec6Aialpha}
\\
\xi_i^\gamma \ =\ & \frac{{{\rm Av}}{{\varepsilon}}\exp A_i^\gamma({{\varepsilon}}) }{\exp A_i^\gamma}.
\label{Sec6xiialpha}\end{aligned}$$ The reason why (\[Sec6Aieps\]) looks different from (\[Aieps\]) is because we fixed Poisson and Gaussian random variables as in (\[fixPoisson\]), (\[fixgs\]), so $\theta^1$ and $\theta_{k}^d$ are defined in terms of $g^1$ and $g_k^d$ in (\[fixgs\]). Again, let us emphasize one more time that all these definition coincide with the old ones when $[p+1]\preceq [p]\gamma$ or, equivalently, when $[1]\preceq \gamma.$
We will keep the dependence on the random variables $(\omega_\beta)_{\beta\preceq [p]},\omega_{[p+1]}, (\omega_\beta^I)_{\beta\preceq [p]}$ implicit and, similarly to (\[Sec5xichi\]), we will write for $\gamma\in{\mathbb{N}}^{r-p}$, $$A_i^{\gamma} =\chi_i \bigl((z_{[p]\beta}^i)_{*\prec \beta \preceq \gamma}\bigr).
\label{Sec6xichi}$$ Let $\chi_{i,r}=\chi_i$ and define for $\gamma\in \Gamma\setminus\{*\}$ by decreasing induction on $|\gamma|$, $$\chi_{i,p+|\gamma|-1}\bigl( (z_{[p]\beta}^i)_{*\prec \beta \prec \gamma} \bigr)
=
\frac{1}{\zeta_{p+|\gamma|-1}} \log {\mathbb{E}}_{z_{[p]\gamma}^i} \exp \zeta_{p+|\gamma|-1} \chi_{i,|\alpha|}\bigl( (z_{[p]\beta}^i)_{*\prec \beta \preceq \gamma}\bigr)$$ and $$W_{p+|\gamma|-1}^i \bigl( (z_{[p]\beta}^i)_{*\prec \beta \preceq \gamma} \bigr)
=
\exp \zeta_{p+|\gamma|-1}\Bigl(\chi_{i,p+|\gamma|}\bigl( (z_{[p]\beta}^i)_{*\prec \beta \preceq \gamma}\bigr) - \chi_{i,p+ |\gamma|-1}\bigl( (z_{[p]\beta}^i)_{*\prec \beta \prec \gamma}\bigr) \Bigr).$$ For $[1]\preceq \gamma$ these are exactly the same definitions as in (\[chij\]) and (\[Sec5Walphai\]), but here we extend these definition to all $\gamma\in \Gamma\setminus \{*\}.$
Let $\tilde{z}_{[p]\beta}^i = (\tilde{\omega}_{[p]\beta}^I)_{I\in{{\cal I}}_i}$ for $\beta \in \Gamma\setminus\{*\}$ be the array generated according to these changes of density along the tree $\Gamma$ as in Section \[Sec2ilabel\]. Since $[p]$ acts as a root, we do not generate any $\tilde{\omega}_{\beta}^I$ for $|\beta|\leq p$. Similarly to (\[Sec4tildas\]), we can write for $\gamma\in{\mathbb{N}}^{r-p}$, $$\xi_i^\gamma = \xi_i \bigl( (z_{[p]\beta}^i)_{*\prec \beta \preceq \gamma}\bigr),\,\,
\tilde{\xi}_i^\gamma = \xi_i \bigl( (\tilde{z}_{[p]\beta}^i)_{*\prec \beta \preceq \gamma} \bigr),
\label{Sec6tildas}$$ where we continue to keep the dependence on $(\omega_\beta)_{\beta\preceq [p]},$ $\omega_{[p+1]}$ and $(\omega_\beta^I)_{\beta\preceq [p]}$, as well as Poisson and Gaussian random variables we fixed above, implicit. Given the vertices $[p+1]\preceq \alpha_1,\ldots,\alpha_k \in {\mathbb{N}}^r$ let $[1] \preceq \gamma_1,\ldots,\gamma_k \in {\mathbb{N}}^{r-p}$ be such that $\alpha_\ell = [p]\gamma_\ell.$ Then we can write $\eta_i^{C}$ in (\[Sec6eqsecond\]) as $$\eta_i^{C} = {{\mathbb{E}_{[p],i}}}\, \xi_i^{\alpha_1}\cdots \xi_i^{\alpha_k} W_{[p],{C}}^{i}
= {\mathbb{E}}_{*,i}\, {{\tilde{\xi}}}_i^{\gamma_1}\cdots {{\tilde{\xi}}}_i^{\gamma_k},
\label{Sec6rep1}$$ where ${\mathbb{E}}_{*,i}$ denotes the expectation in $\tilde{z}_{[p]\beta}^i$ for $\beta \in \Gamma\setminus\{*\}$.
Below we will represent this quantity using the analogue of Theorem \[Th4label\]. Let $(v_{\gamma})_{\gamma\in{\mathbb{N}}^{r-p}}$ be the weights of the Ruelle probability cascades corresponding to the parameters $$0<\zeta_{p}<\ldots<\zeta_{r-1}<1,$$ let $(V_\gamma)_{\gamma\in{\mathbb{N}}^{r-p}}$ be their rearrangement as in (\[Vs2\]) and, similarly to (\[tildeVs\]), define $$\tilde{V}_\gamma
=
\frac{V_\gamma \exp A_i^\gamma}{\sum_{\gamma\in {\mathbb{N}}^{r-p}}V_\gamma \exp A_i^\gamma}.
\label{Sec6tildeVs}$$ Theorem \[Th4label\] can be formulated in this case as follows.
\[Sec6Th4label\] There exists a random bijection $\rho:\Gamma\to\Gamma$ of the vertices of the tree $\Gamma$, which preserves the parent-child relationship, such that $$\bigl(\tilde{V}_{\rho(\gamma)}\bigr)_{\gamma\in{\mathbb{N}}^{r-p}}
\stackrel{d}{=}
\bigl(V_\gamma\bigr)_{\gamma\in{\mathbb{N}}^{r-p}},\,\,
\bigl(z^i_{[p]\rho(\gamma)} \bigr)_{\gamma\in \Gamma\setminus \{*\}}
\stackrel{d}{=}
\bigl(\tilde{z}^i_{[p]\gamma} \bigr)_{\gamma\in \Gamma\setminus \{*\}}
\label{Sec6Th4eq}$$ and these two arrays are independent of each other.
The expectation ${\mathbb{E}}_{*,i}$ in (\[Sec6rep1\]) depends on $\gamma_1,\ldots, \gamma_k$ only through their overlaps $(\gamma_\ell\wedge \gamma_{\ell'})_{\ell,\ell'\leq k}$. Above, we made the specific choice $\alpha_\ell = [p]\gamma_\ell$, which implies $$\gamma_\ell\wedge \gamma_{\ell'} = q_{\ell,\ell'} : =\alpha_\ell\wedge \alpha_{\ell'} - p.$$ Now, consider the set of arbitrary configurations with these overlaps, $${{\cal C}}= \bigl\{ (\gamma_1,\ldots, \gamma_k) \in ({\mathbb{N}}^{r-p})^k \ \bigr|\ \gamma_\ell\wedge \gamma_{\ell'} = q_{\ell,\ell'} \bigr\}.$$ Let us denote by ${\mathbb{E}}_*$ the expectation in $(V_\gamma)_{\gamma\in{\mathbb{N}}^{r-p}}$ in addition to $\tilde{z}_{[p]\beta}^i$ for $\beta \in \Gamma\setminus\{*\}$ in the definition of ${\mathbb{E}}_{*,i}$. We will also denote by ${\mathbb{E}}_*$ the expectation in $(V_\gamma)_{\gamma\in{\mathbb{N}}^{r-p}}$ and $z_{[p]\beta}^i$ for $\beta \in \Gamma\setminus\{*\}$. Using Theorem \[Sec6Th4label\] and arguing as in the proof of Theorem \[Sec4Th\], $$\begin{aligned}
{\mathbb{E}}_*
\sum_{(\gamma_1,\ldots, \gamma_k)\in {{\cal C}}} \tilde{V}_{\gamma_1}\cdots \tilde{V}_{\gamma_k} \,
\xi_i^{\gamma_1}\cdots \xi_i^{\gamma_k}
\ = & \
{\mathbb{E}}_* \sum_{(\gamma_1,\ldots, \gamma_k)\in {{\cal C}}} V_{\gamma_1}\cdots V_{\gamma_k} \,
{\mathbb{E}}_* {{\tilde{\xi}}}_i^{\gamma_1}\cdots {{\tilde{\xi}}}_i^{\gamma_k}
\nonumber
\\
\ = & \
\eta_i^{C}\
{\mathbb{E}}_* \sum_{(\gamma_1,\ldots, \gamma_k)\in {{\cal C}}} V_{\gamma_1}\cdots V_{\gamma_k}.
\label{Sec6repres}\end{aligned}$$ Let us rewrite this equation using a more convenient notation. Let $\sigma^1,\ldots,\sigma^k$ be i.i.d. replicas drawn from ${\mathbb{N}}^{r-p}$ according to the weights $(V_\gamma)_{\gamma\in{\mathbb{N}}^{r-p}}$ and let ${\langle}\,\cdot\, {\rangle}$ denote the average with respect to these weights. If we denote $Q^k = (\sigma^\ell\wedge \sigma^{\ell'})_{\ell,\ell'\leq k}$ and $Q = (q_{\ell,\ell'})_{\ell,\ell'\leq k}$ then we can write $${\mathbb{E}}_* \sum_{(\gamma_1,\ldots, \gamma_k)\in {{\cal C}}} V_{\gamma_1}\cdots V_{\gamma_k}
= {\mathbb{E}}_* \bigl{\langle}{{\rm I}}(Q^k = Q) \bigr{\rangle}= {\mathbb{P}}(Q^k = Q)$$ and $${\mathbb{E}}_*
\sum_{(\gamma_1,\ldots, \gamma_k)\in {{\cal C}}} \tilde{V}_{\gamma_1}\cdots \tilde{V}_{\gamma_k} \,
\xi_i^{\gamma_1}\cdots \xi_i^{\gamma_k}
=
{\mathbb{E}}_*
\frac{\bigl{\langle}\xi_i^{\sigma^1}\cdots \xi_i^{\sigma^k} {{\rm I}}(Q^k = Q) \exp \sum_{\ell\leq k} A_i^{\sigma^\ell} \bigr{\rangle}}{\bigl{\langle}\exp A_i^{\sigma} \bigr{\rangle}^k},$$ and we can rewrite (\[Sec6repres\]) above as $${\mathbb{E}}_*
\frac{\bigl{\langle}\prod_{\ell\leq k} \xi_i^{\sigma^\ell} {{\rm I}}(Q^k = Q) \exp \sum_{\ell\leq k} A_i^{\sigma^\ell} \bigr{\rangle}}{\bigl{\langle}\exp A_i^{\sigma} \bigr{\rangle}^k}
=
\eta_i^{C} {\mathbb{P}}(Q^k = Q).$$ Notice that this computation also works if we replace each factor $\xi_i^{\gamma_\ell}$ in (\[Sec6repres\]) by any power $(\xi_i^{\gamma_\ell})^{n_\ell}$ and, in particular, by setting $n_\ell = 0$ or $1$ we get the following. For a subset $S\subseteq \{1,\ldots, k\}$, let us denote $C(S)=\{\gamma_\ell \ | \ \ell\in S\}.$ Then $${\mathbb{E}}_*
\frac{\bigl{\langle}\prod_{\ell\in S} \xi_i^{\sigma^\ell} {{\rm I}}(Q^k = Q) \exp \sum_{\ell\leq k} A_i^{\sigma^\ell} \bigr{\rangle}}{\bigl{\langle}\exp A_i^{\sigma} \bigr{\rangle}^k}
=
\eta_i^{C(S)} {\mathbb{P}}(Q^k = Q) .$$ Furthermore, it will be convenient to rewrite the left hand side using (\[Sec6Aialpha\]) and (\[Sec6xiialpha\]) as $${\mathbb{E}}_*
\frac{\bigl{\langle}{{\rm Av}}\prod_{\ell\in S} {{\varepsilon}}_\ell \exp \sum_{\ell\leq k} A_i^{\sigma^\ell}({{\varepsilon}}_\ell)\, {{\rm I}}(Q^k = Q) \bigr{\rangle}}{\bigl{\langle}{{\rm Av}}\, \exp \sum_{\ell\leq k} A_i^{\sigma^\ell}({{\varepsilon}}_\ell) \bigr{\rangle}}.$$ We showed as a consequence of the assumption (M) that all $\eta_i^{C(S)}$ do not depend on $\omega_{[p+1]}$ and, therefore, we proved the following.
\[Sec6Thend\] Under the assumption (M), for any subset $S\subseteq \{1,\ldots, k\}$, $${\mathbb{E}}_*
\frac{\bigl{\langle}{{\rm Av}}\prod_{\ell\in S} {{\varepsilon}}_\ell \exp \sum_{\ell\leq k} A_i^{\sigma^\ell}({{\varepsilon}}_\ell)\, {{\rm I}}(Q^k = Q) \bigr{\rangle}}{\bigl{\langle}{{\rm Av}}\, \exp \sum_{\ell\leq k} A_i^{\sigma^\ell}({{\varepsilon}}_\ell) \bigr{\rangle}}
\label{Sec6ThendEq}$$ does not depend on $\omega_{[p+1]}$ almost surely.
Generating Gaussian random fields {#Sec7label}
=================================
We begin by simplifying (\[Sec6Aieps\]) further, by taking $n_2=1$ and setting all other $n_d=0$ for $d\geq 3$ except for one, $n_d = n$, $$A_i^\gamma({{\varepsilon}}) = \theta^1({{\varepsilon}})
+ \theta_{1}^2({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{{1,2,1,i}},{{\varepsilon}})
+\sum_{j\leq n}\theta_{j}^d({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{{1,d,j,i}},\ldots, {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{{d-1,d,j,i}},{{\varepsilon}}).$$ One can easily see that the definition of $\theta^d$ in (\[thetadetx\]) satisfies for ${{\varepsilon}}\in \{-1,+1\},$ $$\theta^d(x_1,\ldots,x_{d-1},{{\varepsilon}}) =\frac{1+{{\varepsilon}}}{2} \log \Bigl(
1+(e^{g^d}-1) \frac{1+x_{1}}{2}\cdots \frac{1+x_{d-1}}{2} \Bigr)$$ and, therefore, we can rewrite $$A_i^\gamma({{\varepsilon}})
=
\frac{1+{{\varepsilon}}}{2}\Bigl(
g^1 + \log\Bigl(1+(e^{g_1^2}-1)\frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{1,2,1,i}}{2}\Bigr)
+\sum_{j\leq n} \log\Bigl(1+(e^{g_j^d}-1)\prod_{\ell\leq d-1}\frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{\ell,d,j,i}}{2}\Bigr)
\Bigr).$$ At this moment, for simplicity of notation, we will drop some unnecessary indices. We will write ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i}$ instead of ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{1,2,1,i}$ and, since $d$ is fixed for a moment, write ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{\ell,j,i}$ instead of ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{\ell,d,j,i}$. Also, we will denote $$x_j:= e^{g_j^d}-1 \in (-1,\infty), \,\,
y: = e^{g_1^2}-1 \in (-1,\infty).$$ Then, we can write $$A_i^\gamma({{\varepsilon}})
=
\frac{1+{{\varepsilon}}}{2}\Bigl(
g^1 + \log\Bigl(1+y \frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i}}{2}\Bigr)
+\sum_{j\leq n} \log\Bigl(1+x_j \prod_{\ell\leq d-1}\frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{\ell,j,i}}{2}\Bigr)
\Bigr).
\label{SecA1}$$ By Theorem \[Sec6Thend\], under the assumption (M), the quantities in (\[Sec6ThendEq\]) do not depend on $\omega_{[p+1]}$ almost surely. In particular, as we discussed above, this almost sure statement can be assumed to hold for all $y,x_j \in (-1,\infty)$ by continuity. We will now take $$x_j = x\frac{\eta_j}{\sqrt{n}}
\label{Sec7etas}$$ for $x\in (-1,1)$ and independent Rademacher random variables $\eta_j$ and show that, by letting $n\to\infty$, we can replace the last sum in (\[SecA1\]) by some Gaussian field in the statement of Theorem \[Sec6Thend\]. Let us denote $$S_{j,i}^\gamma =
\prod_{\ell\leq d-1}\frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{\ell,j,i}}{2}.$$ Then with the choice of $x_j = x\eta_j/\sqrt{n}$ we can use Taylor’s expansion to write $$\sum_{j\leq n} \log\Bigl(1+x_j \prod_{\ell\leq d-1}\frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{\ell,j,i}}{2}\Bigr)
=
\frac{x}{\sqrt{n}} \sum_{j\leq n} \eta_j S_{j,i}^\gamma
- \frac{x^2}{2n} \sum_{j\leq n} (S_{j,i}^\gamma)^2
+ O(n^{-1/2}).
\label{Sec7napprox}$$ The last term $O(n^{-1/2})$ is uniform in all parameters, so it will disappear in (\[Sec6ThendEq\]) when we let $n$ go to infinity. For the first term, we will use the classical CLT to replace it by Gaussian and for the second term we will use the SLLN, which will produce a term that will cancel out in the numerator and denominator in (\[Sec6ThendEq\]). However, before we do that, we will need to change the definition of the expectation ${\mathbb{E}}_*$ slightly. Recall that, by (\[Sec6sialpha\]), $${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{\ell, j,i}
=
h\bigl( (\omega_\beta)_{\beta \preceq [p]}, \omega_{[p+1]}, (\omega_\beta)_{\beta \preceq [p]}^{\ell, j,i} , (\omega_{[p]\beta}^{\ell, j,i} )_{*\prec \beta \preceq \gamma} \bigr).$$ In (\[Sec6ThendEq\]) we already average in the random variables $\omega_{[p]\beta}^{\ell, j,i}$ for $*\prec \beta \preceq \gamma$ but, clearly, the statement of Theorem \[Sec6Thend\] holds if ${\mathbb{E}}_*$ also includes the average in $(\omega_\beta^{\ell, j,i})_{\beta\preceq [p]}$ and the Rademacher random variables $\eta_j$ in (\[Sec7etas\]). From now on we assume this. Note that ${\mathbb{E}}_*$ still does not include the average with respect to the random variables $(\omega_\beta^i)_{\beta\preceq [p]}$ that appear in ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\gamma$ in (\[SecA1\]), $${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i}
=
h\bigl( (\omega_\beta)_{\beta\preceq [p]}, \omega_{[p+1]}, (\omega_\beta^i)_{\beta\preceq [p]}, (\omega_{[p]\beta}^{i} )_{*\prec \beta \preceq \gamma} \bigr).
\label{Sec7sbar}$$ Of course, one can not apply the CLT and SLLN in (\[Sec6ThendEq\]) directly, because there are infinitely many terms indexed by $\gamma\in{\mathbb{N}}^{r-p}$. However, this is not a serious problem because most of the weight of the Ruelle probability cascades $(V_\gamma)$ is concentrated on finitely many indices $\gamma$ and it is not difficult to show that (\[Sec6ThendEq\]) is well approximated by the analogous quantity where the series over $\gamma$ are truncated at finitely many terms. Moreover, this approximation is uniform over $n$ in (\[Sec7napprox\]). This is why the representation of $\eta_i^{C(S)}$ in the previous section using the Ruelle probability cascades plays such a crucial role. We will postpone the details until later in this section and first explain what happens in (\[Sec7napprox\]) for finitely many $\gamma$.
First of all, by the SLLN, for any fixed $(\omega_\beta)_{\beta\preceq [p]}$ and $\omega_{[p+1]}$, $$\lim_{n\to\infty}\frac{1}{n} \sum_{j\leq n} (S_{j,i}^\gamma)^2
= {\mathbb{E}}_* (S_{1,i}^\gamma)^2
= \prod_{\ell\leq d-1} {\mathbb{E}}_* \Bigl(\frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{\ell,1,i}}{2}\Bigr)^2
= \Bigl({\mathbb{E}}_* \Bigl(\frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{1,1,i}}{2}\Bigr)^2\Bigr)^{d-1}$$ almost surely. Of course, we can now simplify the notation by replacing ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{1,1,i}$ with ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i}$ and replacing ${\mathbb{E}}_*$ by the expectation ${\mathbb{E}}_i$ with respect to $\omega_\beta^i$ for $\beta\in{{\cal A}}$, $$\lim_{n\to\infty}\frac{1}{n} \sum_{j\leq n} (S_{j,i}^\gamma)^2
= \Bigl({\mathbb{E}}_i \Bigl(\frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i}}{2}\Bigr)^2\Bigr)^{d-1}
= 2^{2-2d}\Bigl( 1+2{\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i}+{\mathbb{E}}_i ({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i})^2 \Bigr)^{d-1}.$$ Lemma \[Sec2iLem1\] in Section \[Sec2ilabel\] (for $p=0$ there) yields that ${\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i}$ depends only on $\omega_*$, $$q_0(\omega_*): = {\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i},
\label{Sec7q0}$$ and, since ${\mathbb{E}}_i ({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i})^2$ clearly does not depend on $\gamma$, we can take $[1]\preceq \gamma$, in which case $${\mathbb{E}}_i ({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i})^2 = {\mathbb{E}}_i ({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{[p]\gamma}_{i})^2 = R_{[p]\gamma,[p]\gamma} = q_r.$$ This means that for any $\gamma\in {\mathbb{N}}^{r-p}$, $$\lim_{n\to\infty}\frac{1}{n} \sum_{j\leq n} (S_{j,i}^\gamma)^2 = 2^{2-2d}\bigl(1+2q_0(\omega_*)+q_r \bigr)^{d-1}
\label{Sec7LLN}$$ almost surely. So, in the limit, these terms will cancel out in (\[Sec6ThendEq\]) – at least when we truncate the summation over $\gamma$ to finitely many $\gamma$, as we shall do below.
Next, let us look at the first sum in (\[Sec7napprox\]) for $\gamma\in F$ for a finite set $F\subset {\mathbb{N}}^{r-p}$. By the classical multivariate CLT (applied for a fixed $(\omega_\beta)_{\beta\preceq [p]}$ and $\omega_{[p+1]}$), $$\frac{1}{\sqrt{n}} \sum_{j\leq n} \eta_k \bigl(S_{j,i}^\gamma\bigr)_{\gamma\in F}
\stackrel{d}{\longrightarrow}
(g^\gamma)_{\gamma\in F},
\label{Sec7CLT}$$ where $(g^\gamma)_{\gamma\in F}$ is a centered Gaussian random vector with the covariance $${\mathbb{E}}g^\gamma g^{\gamma'}
=
{\mathbb{E}}_* S_{1,i}^\gamma S_{1,i}^{\gamma'}
=
\Bigl(
{\mathbb{E}}_i \frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\gamma}{2}\cdot\frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\gamma'}}{2}
\Bigr)^{d-1}
=
\Bigl( \frac{1}{4}\bigl(1+{\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i}+{\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\gamma'}_{i}+{\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\gamma'}_{i} \bigr)\Bigr)^{d-1}.$$ First of all, as above ${\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i} = {\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\gamma'}_{i} = q_0(\omega_*).$ To compute ${\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\gamma'}_{i}$, we need to consider two cases. First, suppose that $\gamma\wedge\gamma'\geq 1.$ Since ${\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\gamma'}_{i}$ clearly depends only on $\gamma\wedge\gamma'$, we can suppose that $[1]\preceq \gamma,\gamma'$, in which case ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\gamma}_{i} = {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{[p]\gamma}_{i}$, ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\gamma'}_{i} = {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{[p]\gamma'}_{i}$ and $${\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\gamma'}_{i}
=
{\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{[p]\gamma}_{i} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{[p]\gamma'}_{i}
= R_{[p]\gamma,[p]\gamma'} = q_{p+\gamma\wedge\gamma'}.$$ In the second case, $\gamma\wedge\gamma' = 0$, when computing ${\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\gamma'}_{i}$ we can first average ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\gamma$ with respect to $ (\omega_{[p]\beta}^{i} )_{*\prec \beta \preceq \gamma}$ and ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\gamma'}$ with respect to $ (\omega_{[p]\beta}^{i} )_{*\prec \beta \preceq \gamma'}$, since these are independent. However, by Lemma \[Sec2iLem1\], both of these averages do not depend on $\omega_{[p+1]}$. This means that, after taking these averages, we can replace $\omega_{[p+1]}$ in (\[Sec7sbar\]) by $\omega_{[p+j]}$ if $[j]\preceq \gamma$, and the same for $\gamma'$. As a result, we can again write $${\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
=
= R_{[p]\gamma,[p]\gamma'} = q_p = q_{p+\gamma\wedge\gamma'}$$ almost surely. If we introduce the notation $$c_{j}(\omega_*) = \frac{1+2q_0(\omega_*) + q_{p+j}}{4}
\,\,\mbox{ for }\,\,
j=0,\ldots, r-p$$ then we proved that the covariance is given by $${\mathbb{E}}g^\gamma g^{\gamma'}
=
{\mathbb{E}}_* S_{1,i}^\gamma S_{1,i}^{\gamma'}
=
c_{\gamma\wedge\gamma'}(\omega_*)^{d-1}.
\label{Sec7cov}$$ Let us show right away that $$0\leq c_0(\omega_*) < \ldots < c_{r-p}(\omega_*).
\label{Sec7cs}$$ In particular, this means that the covariance in (\[Sec7cov\]) is increasing with $\gamma\wedge\gamma'$ and the Gaussian field $(g^\gamma)_{\gamma\in{\mathbb{N}}^{r-p}}$ is the familiar field that accompanies the Ruelle probability cascades in the pure $d$-spin non-diluted models. Of course, the only statement that requires a proof is the following.
The inequality $c_0(\omega_*)\geq 0$ holds almost surely.
**Proof.** First of all, let us notice that, by Lemma \[Sec2iLem1\], we can also write the definition of $q_0(\omega_*)$ in (\[Sec7q0\]) as $
q_0(\omega_*) = {\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\alpha_{i}
$ for any vertex $\alpha\in{\mathbb{N}}^r$, where ${\mathbb{E}}_i$ denotes the expectation in the random variables $\omega_\beta^i$ for $\beta\in{{\cal A}}.$ Let ${\mathbb{E}}_i'$ denote the expectation with respect to $\omega_\beta^i$ for $\beta\in{{\cal A}}\setminus \{*\}$, excluding the average in $\omega_*^i$. Again, by Lemma \[Sec2iLem1\], we can write ${\mathbb{E}}_i' {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\alpha_{i}$ as $q_0(\omega_*,\omega_*^i)$. If we take any $\alpha,\alpha'\in {\mathbb{N}}^{r}$ such that $\alpha\wedge\alpha'=0$ then $$q_0(\omega_*)^2 = ({\mathbb{E}}_i {\mathbb{E}}_i'{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\alpha_{i})^2
\leq {\mathbb{E}}_i ({\mathbb{E}}_i'{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\alpha_{i})^2 = {\mathbb{E}}_i ({\mathbb{E}}_i'{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\alpha_{i} {\mathbb{E}}_i'{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\alpha'}_{i})
=
{\mathbb{E}}_i {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\alpha_{i} {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^{\alpha'}_{i}
=q_{\alpha\wedge\alpha'} = q_0$$ almost surely. Therefore $$4c_0(\omega_*)\geq 1-2\sqrt{q_0} +q_0 = (1-\sqrt{q_0})^2\geq 0.$$ This finishes the proof.
From now on let ${\mathbb{E}}_*$ also include the expectation in the Gaussian field $(g^\gamma)_{\gamma\in{\mathbb{N}}^{r-p}}$ (conditionally on $\omega_*$) with the covariance (\[Sec7cov\]). Let us denote by $$a_i^\gamma({{\varepsilon}})
=
\frac{1+{{\varepsilon}}}{2}\Bigl(
g^1 + \log\Bigl(1+y \frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i}}{2}\Bigr)
+x g^\gamma
\Bigr)
\label{Sec7a1}$$ the quantity that will replace $A_i^\gamma({{\varepsilon}})$ in (\[SecA1\]) in the limit $n\to\infty$. We are ready to prove the following.
\[Sec7Th\] Under the assumption (M), for any subset $S\subseteq \{1,\ldots, k\}$, $${\mathbb{E}}_*
\frac{\bigl{\langle}{{\rm Av}}\prod_{\ell\in S} {{\varepsilon}}_\ell \exp \sum_{\ell\leq k} a_i^{\sigma^\ell}({{\varepsilon}}_\ell)\, {{\rm I}}(Q^k = Q) \bigr{\rangle}}{\bigl{\langle}{{\rm Av}}\, \exp \sum_{\ell\leq k} a_i^{\sigma^\ell}({{\varepsilon}}_\ell) \bigr{\rangle}}
\label{Sec7ThEq}$$ does not depend on $\omega_{[p+1]}$ almost surely.
**Proof.** We only need to show that the quantity in (\[Sec6ThendEq\]) with $A_i^\gamma({{\varepsilon}})$ as in (\[SecA1\]) with the choice of $x_j$ as in (\[Sec7etas\]) converges to (\[Sec7ThEq\]), where ${\mathbb{E}}_*$ was redefined above (\[Sec7sbar\]) to include the average over $(\omega_\beta)_{\beta\preceq [p]}^{\ell, j,i}$ and Rademacher random variables $\eta_j$ in (\[Sec7etas\]), as well as the Gaussian field $(g^\gamma)_{\gamma\in{\mathbb{N}}^{r-p}}$ with the covariance (\[Sec7cov\]).
Let $(V_m')_{m\geq 1}$ be the RPC weights $(V_\gamma)_{\gamma\in{\mathbb{N}}^{r-p}}$ arranged in the decreasing order. For some fixed large $M\geq 1$, let us separate the averages ${\langle}\, \cdot\, {\rangle}$ in the numerator and denominator in (\[Sec6ThendEq\]) into two sums over $V_m'$ for $m\leq M$ and for $m>M$ (for each replica). Let us denote the corresponding averages by ${\langle}\, \cdot\, {\rangle}_{\leq M}$ and ${\langle}\, \cdot\, {\rangle}_{>M}$. Let $$\begin{aligned}
a \ =&\
\bigl{\langle}{{\rm Av}}\prod_{\ell\in S} {{\varepsilon}}_\ell \exp \sum_{\ell\leq k} A_i^{\sigma^\ell}({{\varepsilon}}_\ell)\, {{\rm I}}(Q^k = Q) \bigr{\rangle}_{\leq M},
\\
b \ =&\
\bigl{\langle}{{\rm Av}}\prod_{\ell\in S} {{\varepsilon}}_\ell \exp \sum_{\ell\leq k} A_i^{\sigma^\ell}({{\varepsilon}}_\ell)\, {{\rm I}}(Q^k = Q) \bigr{\rangle}_{> M},
\\
c \ =&\
\bigl{\langle}{{\rm Av}}\, \exp \sum_{\ell\leq k} A_i^{\sigma^\ell}({{\varepsilon}}_\ell) \bigr{\rangle}_{\leq M},
\\
d \ =&\
\bigl{\langle}{{\rm Av}}\, \exp \sum_{\ell\leq k} A_i^{\sigma^\ell}({{\varepsilon}}_\ell) \bigr{\rangle}_{> M}.\end{aligned}$$ Note that $|a|\leq c$, $|b|\leq d$ and $$\Bigl|\frac{a+b}{c+d} - \frac{a}{c}\Bigr|
=
\Bigl|\frac{bc-ad}{c(c+d)}\Bigr|
\leq
\Bigl|\frac{b}{c+d}\Bigr|+\Bigl|\frac{d}{c+d}\Bigr|
\leq
\frac{2d}{c+d}.$$ This means that the difference between (\[Sec6ThendEq\]) and $${\mathbb{E}}_* \frac{a}{c}
=
{\mathbb{E}}_*
\frac{\bigl{\langle}{{\rm Av}}\prod_{\ell\in S} {{\varepsilon}}_\ell \exp \sum_{\ell\leq k} A_i^{\sigma^\ell}({{\varepsilon}}_\ell)\, {{\rm I}}(Q^k = Q) \bigr{\rangle}_{\leq M}}{\bigl{\langle}{{\rm Av}}\, \exp \sum_{\ell\leq k} A_i^{\sigma^\ell}({{\varepsilon}}_\ell) \bigr{\rangle}_{\leq M}}
\label{Sec7trunc}$$ can be bounded by $2{\mathbb{E}}_* d/(c+d)$. By (\[Sec7napprox\]), the SLLN in (\[Sec7LLN\]) and CLT in (\[Sec7CLT\]), with probability one, (\[Sec7trunc\]) converges to $${\mathbb{E}}_*
\frac{\bigl{\langle}{{\rm Av}}\prod_{\ell\in S} {{\varepsilon}}_\ell \exp \sum_{\ell\leq k} a_i^{\sigma^\ell}({{\varepsilon}}_\ell)\, {{\rm I}}(Q^k = Q) \bigr{\rangle}_{\leq M}}{\bigl{\langle}{{\rm Av}}\, \exp \sum_{\ell\leq k} a_i^{\sigma^\ell}({{\varepsilon}}_\ell) \bigr{\rangle}_{\leq M}}
\label{Sec7limM}$$ as $n\to\infty$. Next, we show that ${\mathbb{E}}_* d/(c+d)$ is small for large $M$, uniformly over $n$. Since $d/(c+d)\in [0,1]$, it is enough to show that $d$ is small and $c$ is not too small with high probability. To show that $d$ is small, we will use Chebyshev’s inequality and show that ${\mathbb{E}}_* d$ is small. By Jensen’s inequality, $$d =
\bigl{\langle}{{\rm Av}}\, \exp \sum_{\ell\leq k} A_i^{\sigma^\ell}({{\varepsilon}}_\ell) \bigr{\rangle}_{> M}
=
\bigl{\langle}{{\rm Av}}\, \exp A_i^{\sigma}({{\varepsilon}}) \bigr{\rangle}_{> M}^k
\leq
\bigl{\langle}{{\rm Av}}\, \exp k\, A_i^{\sigma}({{\varepsilon}}) \bigr{\rangle}_{> M}.$$ If we denote $\delta_M = {\mathbb{E}}\sum_{m>M} V_m'$ then, since the weights $(V_\gamma)$ and the random variables in $A_i^\gamma({{\varepsilon}})$ are independent, $${\mathbb{E}}_*d \leq
\delta_M
\sup_{\gamma} {{\rm Av}}\, {\mathbb{E}}_* \exp k\, A_i^{\gamma}({{\varepsilon}}).$$ Using that $\log(1+t)\leq t$, we can bound $A_i^{\gamma}({{\varepsilon}})$ with the choice (\[Sec7etas\]) by $$A_i^{\gamma}({{\varepsilon}})
\leq |g^1| + \log\bigl(1+|y|\bigr) + \frac{1+{{\varepsilon}}}{2}\frac{x}{\sqrt{n}} \sum_{j\leq n} \eta_j S_{j,i}^\gamma.$$ Using that, for a Rademacher random variable $\eta$, we have ${\mathbb{E}}e^{\eta t} = {{\mbox{\rm ch}}}(t) \leq e^{t^2/2}$ we get that $${\mathbb{E}}_\eta \exp k\frac{1+{{\varepsilon}}}{2}\frac{x}{\sqrt{n}} \sum_{j\leq n} \eta_j S_{j,i}^\gamma
\leq e^{k^2/2}$$ and, therefore, $$\sup_{\gamma} {{\rm Av}}\, {\mathbb{E}}_* \exp k\, A_i^{\gamma}({{\varepsilon}})
\leq
c_k:=\exp\Bigl( k|g^1| + k\log\bigl(1+|y|\bigr) +\frac{k^2}{2} \Bigr).$$ We showed that ${\mathbb{E}}_* d \leq c_k \delta_M$, and this bound does not depend on $n$. Since $\delta_M$ is small for $M$ large, $d$ is small with high probability uniformly over $n.$ On the other hand, to show that $c$ is not too small, we simply bound it from below by $$c\geq \bigl(V_\gamma\, {{\rm Av}}\exp A_i^{\gamma}({{\varepsilon}})\bigr)^k$$ for $\gamma$ corresponding to the largest weight, $V_1' = V_\gamma.$ The weight $V_1'$ is strictly positive and its distribution does not depend on $n$. Also, using (\[Sec7napprox\]), we can bound $A_i^{\gamma}({{\varepsilon}})$ from below by $$A_i^{\gamma}({{\varepsilon}})
\geq
-|g^1| + \log\bigl(1-|y|\bigr) - \Bigl|\frac{1}{\sqrt{n}} \sum_{j\leq n} \eta_j S_{j,i}^\gamma \Bigr| - L$$ for some absolute constant $L$. Even though the index $\gamma$ here is random, because it corresponds to the largest weight $V_1'$, we can control this quantity using Hoeffding’s inequality for Rademacher random variables conditionally on other random variables to get $${\mathbb{P}}\Bigl(
\Bigl|\frac{1}{\sqrt{n}} \sum_{j\leq n} \eta_j S_{j,i}^\gamma \Bigr| \geq t
\Bigr)
\leq 2e^{-t^2/2}.$$ Therefore, for any $\delta>0$ there exists $\Delta>0$ (that depends on $|g^1|$, $|y|$, $k$ and the distribution of $V_1'$) such that ${\mathbb{P}}(c\geq \Delta)\geq 1-\delta.$ All together, we showed that ${\mathbb{E}}_* d/(c+d)$ is small for large $M$, uniformly over $n$.
To finish the proof, we need to show that (\[Sec7limM\]) approximates (\[Sec7ThEq\]) for large $M$. Clearly, this can be done by the same argument (only easier) that we used above to show that (\[Sec7trunc\]) approximates (\[Sec6ThendEq\]).
Final consequences of the cavity equations
==========================================
Theorem \[Sec7Th\] implies that, under the assumption (M), $${\mathbb{E}}_*
\frac{\bigl{\langle}{{\rm Av}}\prod_{\ell\leq k}(1+ {{\varepsilon}}_\ell) \exp \sum_{\ell\leq k} a_i^{\sigma^\ell}({{\varepsilon}}_\ell)\, {{\rm I}}(Q^k = Q) \bigr{\rangle}}{\bigl{\langle}{{\rm Av}}\, \exp \sum_{\ell\leq k} a_i^{\sigma^\ell}({{\varepsilon}}_\ell) \bigr{\rangle}}
\label{Sec8Eq1}$$ does not depend on $\omega_{[p+1]}$ almost surely. This follows from (\[Sec7ThEq\]) by multiplying out $\prod_{\ell\leq k}(1+ {{\varepsilon}}_\ell).$ Using that for ${{\varepsilon}}\in\{-1,+1\}$, $$\exp t \frac{1+{{\varepsilon}}}{2} = 1+ (e^t-1)\frac{1+{{\varepsilon}}}{2},$$ one can see from (\[Sec7a1\]) that $$2{{\rm Av}}\exp
a_i^\gamma({{\varepsilon}})
=
1+e^{g^1+xg^\gamma}
\Bigl(1+y \frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i}}{2}\Bigr)$$ and$${{\rm Av}}(1+{{\varepsilon}}) \exp a_i^\gamma({{\varepsilon}})
=
e^{g^1+xg^\gamma}
\Bigl(1+y \frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i}}{2}\Bigr).
\medskip$$ If for simplicity the notation we denote $z:=e^{g^1}\in(0,\infty)$ and $$S_i^\gamma = \frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\gamma_{i}}{2}
\label{Sec8Sgamma}$$ then (\[Sec8Eq1\]) can be written, up to a factor $2^k z^k$, as $${\mathbb{E}}_*
\frac{\bigl{\langle}\prod_{\ell\leq k} \exp(xg^{\sigma^\ell})(1+yS_i^{\sigma^\ell})\, {{\rm I}}(Q^k = Q) \bigr{\rangle}}{\bigl{\langle}1+z \exp(xg^{\sigma})(1+yS_i^{\sigma}) \bigr{\rangle}^k}.$$ As before, the statement that this quantity does not depend on $\omega_{[p+1]}$ almost surely holds for all $x\in (-1,1)$, $y>-1$ and $z>0$, by continuity. Therefore, if we take the derivative with respect to $z$ and then let $z\downarrow 0$ then the quantity we get (up to a factor $-k$), $${\mathbb{E}}_* \Bigl{\langle}\prod_{\ell\leq k+1} \exp(xg^{\sigma^\ell})(1+yS_i^{\sigma^\ell})\, {{\rm I}}(Q^k = Q) \Bigr{\rangle},$$ does not depend on $\omega_{[p+1]}$ almost surely. This is a polynomial in $y$ of order $k+1$ and if we take the derivative $\partial^{k+1}/\partial y^{k+1}$ we get that $${\mathbb{E}}_* \Bigl{\langle}\prod_{\ell\leq k+1} \exp(x g^{\sigma^\ell}) \,S_i^{\sigma^\ell} \, {{\rm I}}(Q^k = Q) \Bigr{\rangle},$$ does not depend on $\omega_{[p+1]}$ almost surely. Let us now take the expectation ${\mathbb{E}}_g$ with respect to the Gaussian field $(g^\gamma)$. By (\[Sec7cov\]), $${\mathbb{E}}_g \exp x\sum_{\ell\leq k+1} g^{\sigma^\ell}
=
\exp \frac{x^2}{2}\sum_{\ell,\ell'\leq k+1} c_{\sigma^\ell\wedge \sigma^{\ell'}}(\omega_*)^{d-1}$$ and, therefore, the quantity $${\mathbb{E}}_* \Bigl{\langle}\prod_{\ell\leq k+1} S_i^{\sigma^\ell} \, {{\rm I}}(Q^k = Q)
\exp \frac{x^2}{2}\sum_{\ell,\ell'\leq k+1} c_{\sigma^\ell\wedge \sigma^{\ell'}}(\omega_*)^{d-1}
\Bigr{\rangle}$$ does not depend on $\omega_{[p+1]}$ almost surely. Notice that because of the indicator ${{\rm I}}(Q^k = Q)$, all the overlaps $\sigma^\ell\wedge\sigma^{\ell'}$ are fixed for $\ell,\ell' \leq k$, so the factor $$\exp \frac{x^2}{2}\sum_{\ell,\ell'\leq k} c_{\sigma^\ell\wedge \sigma^{\ell'}}(\omega_*)^{d-1}$$ can be taken outside of ${\mathbb{E}}_* {\langle}\,\cdot \,{\rangle}$ and cancelled out, yielding that $${\mathbb{E}}_* \Bigl{\langle}\prod_{\ell\leq k+1} S_i^{\sigma^\ell} \, {{\rm I}}(Q^k = Q)
\exp \frac{x^2}{2}\sum_{\ell\leq k} c_{\sigma^\ell\wedge \sigma^{k+1}}(\omega_*)^{d-1}
\Bigr{\rangle}$$ does not depend on $\omega_{[p+1]}$ almost surely. Taking the derivative with respect to $x^2/2$ at zero gives that $${\mathbb{E}}_* \Bigl{\langle}\prod_{\ell\leq k+1} S_i^{\sigma^\ell} \, {{\rm I}}(Q^k = Q)
\sum_{\ell \leq k} c_{\sigma^\ell\wedge \sigma^{k+1}}(\omega_*)^{d-1}
\Bigr{\rangle}\label{Sec9polyd}$$ does not depend on $\omega_{[p+1]}$ almost surely. We proved this statement for a fixed but arbitrary $d\geq 3$, but it also holds for $d=1$ by setting $x=0$ in the previous equation. Let us take arbitrary $f_j$ for $j=0,\ldots,r-p$ and consider a continuous function $f$ on $[0,1]$ such that $f(c_j(\omega_*)) = f_j.$ Approximating this function by polynomials, (\[Sec9polyd\]) implies that $${\mathbb{E}}_* \Bigl{\langle}\prod_{\ell\leq k+1} S_i^{\sigma^\ell} \, {{\rm I}}(Q^k = Q)
\sum_{\ell \leq k} \sum_{j=0}^{r-p}f_j {{\rm I}}(\sigma^\ell\wedge \sigma^{k+1}=j)
\Bigr{\rangle}$$ does not depend on $\omega_{[p+1]}$ for all $(f_j)$ almost surely. Taking the derivative in $f_j$ shows that $${\mathbb{E}}_* \Bigl{\langle}\prod_{\ell\leq k+1} S_i^{\sigma^\ell} \, {{\rm I}}(Q^k = Q)
\sum_{\ell \leq k} {{\rm I}}(\sigma^\ell\wedge \sigma^{k+1}=j)
\Bigr{\rangle}$$ does not depend on $\omega_{[p+1]}$ almost surely and, therefore, $${\mathbb{E}}_* \Bigl{\langle}\prod_{\ell\leq k+1} S_i^{\sigma^\ell} \, {{\rm I}}(Q^k = Q)
\sum_{\ell \leq k} {{\rm I}}(\sigma^\ell\wedge \sigma^{k+1}\geq j)
\Bigr{\rangle}$$ does not depend on $\omega_{[p+1]}$ almost surely for all $j=0,\ldots, r-p$. Let us now express this quantity as a linear combination over all possible overlap configurations that the new replica $\sigma^{k+1}$ can form with the old replicas $\sigma^1,\ldots,\sigma^k$.
Given a $k\times k$ overlap constraint matrix $Q=(q_{\ell,\ell'})_{\ell,\ell'\leq k}$, let ${{\cal E}}(Q)$ be the set of admissible extensions $Q'=(q_{\ell,\ell'}')_{\ell,\ell'\leq k+1}$ of $Q$ to $(k+1)\times(k+1)$ overlap constraint matrices. In other words, $q_{\ell,\ell'}' = q_{\ell,\ell'}$ for $\ell,\ell'\leq k$, and there exists $\gamma_1,\ldots,\gamma_{k+1}\in {\mathbb{N}}^{r-p}$ such that $\gamma_\ell\wedge \gamma_{\ell'} = q_{\ell,\ell'}'$ for $\ell,\ell'\leq k+1.$ If we denote $$n_j(Q') = \sum_{\ell \leq k} {{\rm I}}(q_{\ell,k+1}'\geq j)
\label{Sec8nj}$$ and denote $Q^{k+1} = (\sigma^\ell\wedge\sigma^{\ell'})_{\ell,\ell'\leq k+1}$ then we showed that $$\sum_{Q'\in{{\cal E}}(Q)} n_j(Q')\,
{\mathbb{E}}_* \Bigl{\langle}\prod_{\ell\leq k+1} S_i^{\sigma^\ell} \, {{\rm I}}(Q^{k+1} = Q')
\Bigr{\rangle}$$ does not depend on $\omega_{[p+1]}$ almost surely for all $j=0,\ldots, r-p$. Since the RPC weights $(V_\gamma)$ are independent of $({{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\gamma)$, we can rewrite this as $$\sum_{Q'\in{{\cal E}}(Q)} n_j(Q') {\mathbb{P}}(Q^{k+1} = Q')\, M(Q')
\label{Sec8at}$$ does not depend on $\omega_{[p+1]}$ almost surely for all $j=0,\ldots, r-p$, where $$M(Q') = {\mathbb{E}}_* \prod_{\ell\leq k+1} S_i^{\gamma_\ell}$$ for any $\gamma_1,\ldots,\gamma_{k+1}\in {\mathbb{N}}^{r-p}$ such that $\gamma_\ell\wedge \gamma_{\ell'} = q_{\ell,\ell'}'$ for $\ell,\ell'\leq k+1.$
Recall that this statement was proved under the induction assumption (M) in Section \[Sec2ilabel\], so let us express (\[Sec8at\]) in the notation of Section \[Sec2ilabel\] and connect everything back to the assumption (M), which we repeat one more time. Given $[p+1]\preceq \alpha_1,\ldots,\alpha_k \in {\mathbb{N}}^r$, we assumed that:
1. for any subset $S\subseteq \{1,\ldots, k\}$, ${{\mathbb{E}_{[p],i}}}\prod_{\ell\in S}{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}$ does not depend on $\omega_{[p+1]}$.
If similarly to (\[Sec8Sgamma\]) we denote, for $\alpha\in{\mathbb{N}}^r$, $$S_i^\alpha = \frac{1+{{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\alpha_{i}}{2}
\label{Sec8Salpha}$$ then the assumption (M) is, obviously, equivalent to
1. for any subset $S\subseteq \{1,\ldots, k\}$, ${{\mathbb{E}_{[p],i}}}\prod_{\ell\in S}S_i^{\alpha_\ell}$ does not depend on $\omega_{[p+1]}$.
Let us define $[1] \preceq \gamma_1,\ldots,\gamma_{k}\in {\mathbb{N}}^{r-p}$ by $\alpha_\ell = [p]\gamma_\ell$, and let $Q$ be the overlap matrix $$q_{\ell,\ell'} = \gamma^\ell\wedge \gamma^{\ell'}
\label{Sec8Qk}$$ The assumption (M) depends on $\alpha_1,\ldots,\alpha_k$ only through this matrix $Q$, so one should really view it as a statement about such $Q$. Fix $1\leq j \leq r-p$ and consider any $Q'\in{{\cal E}}(Q)$ such that $n_j(Q')\not = 0.$ Since $$n_j(Q') \not = 0 \Longleftrightarrow \max_{\ell\leq k} q_{\ell,k+1}'\geq j,$$ this means that $q_{\ell,k+1}' \geq j\geq 1$ for some $\ell\leq k$. Since our choice of $\alpha_1,\ldots,\alpha_k \in {\mathbb{N}}^r$ was such that $q_{\ell,\ell'} = \gamma_\ell\wedge \gamma_{\ell'}\geq 1$ for all $\ell,\ell'\leq k$, this also implies that $q_{\ell,\ell'}'\geq 1$ for all $\ell,\ell'\leq k+1$. In particular, we can find $[1] \preceq \gamma_{k+1}\in {\mathbb{N}}^{r-p}$ such that $\gamma_\ell\wedge \gamma_{k+1} = q_{\ell,k+1}'$ for $\ell\leq k$. Let $\alpha_{k+1} = [p]\gamma_{k+1}.$ Recall that, whenever $\alpha = [p]\gamma$ for $[1] \preceq \gamma\in {\mathbb{N}}^{r-p}$, the definitions (\[Sec6sialpha\]) and (\[indass\]) imply that ${{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}^\alpha_i = {{{ \sbox{\myboxA}{$\m@ths$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.75\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}\hspace{0.1mm}}}_i^\gamma$. Therefore, in this case, we can rewrite the definition of $M(Q')$ below (\[Sec8at\]) as $$M(Q') = {\mathbb{E}}_* \prod_{\ell\leq k+1} S_i^{\gamma_\ell}
= {{\mathbb{E}_{[p],i}}}\prod_{\ell\leq k+1} S_i^{\alpha_\ell}.$$ Let us summarize what we proved.
\[Sec8Th\] If the matrix $Q$ defined in (\[Sec8Qk\]) satisfies the assumption (M) then $$\sum_{Q'\in{{\cal E}}(Q)} n_j(Q') {\mathbb{P}}(Q^{k+1} = Q')\, M(Q')
\label{Sec8ThEq}$$ does not depend on $\omega_{[p+1]}$ almost surely for all $j=1,\ldots, r-p$.
Main induction argument {#Sec9label}
=======================
Finally, we will now use Theorem \[Sec8Th\] to prove our main goal, Theorem \[Sec6iTh1\] in Section \[Sec2ilabel\]. To emphasize that our inductive proof will have a monotonicity property (M), we can rephrase Theorem \[Sec6iTh1\] as follows.
\[Sec9Th1\] Under the assumption (\[indass\]), for any $k\geq 1$, any $[p+1]\preceq \alpha_1,\ldots,\alpha_k \in {\mathbb{N}}^r$ and any $S\subseteq \{1,\ldots, k\}$, the expectation ${{\mathbb{E}_{[p],i}}}\prod_{\ell\in S} S_i^{\alpha_\ell}$ with respect to $(\omega_\beta^i)_{[p+1]\preceq \beta}$ does not depend on $\omega_{[p+1]}$ almost surely.
It is much easier to describe the proof if we represent a configuration $[p+1]\preceq \alpha_1,\ldots,\alpha_k \in {\mathbb{N}}^r$ not by a matrix $Q = (\gamma_\ell\wedge\gamma_{\ell'})$ with $\alpha_\ell = [p]\gamma_\ell$ but by a subtree of ${{\cal A}}$ growing out of the vertex $[p+1]$ with branches leading to the leaves $\alpha_1,\ldots,\alpha_k$, and with an additional layer encoding their multiplicities (see Fig. \[Fig1\]). If we think of $[p+1]$ as a root of this subtree being at depth zero, then the leaves are at depth $r-p-1.$ However, the multiplicity of any particular vertex $\alpha$ in the set $\{\alpha_1,\ldots,\alpha_k\}$ can be greater than one, so we will attach that number of children to each vertex $\alpha$ to represent multiplicities, so the depth of the tree will be $r-p$. Whenever we say that we remove a leaf $\alpha$ from the tree, we mean that we remove one multiplicity of $\alpha$. Notice that removing a leaf from the tree also removes the path to that leaf, of course, keeping the shared paths leading to other leaves that are still there.
We will say that this *tree is good* if $$M(Q) = {{\mathbb{E}_{[p],i}}}\prod_{\ell\leq k} S_i^{\alpha_\ell}$$ does not depend on $\omega_{[p+1]}$.
![\[Fig1\] Representing a configuration $\alpha_1,\ldots,\alpha_k$ by a tree.](figure1.eps){width="45.00000%"}
First we are going to prove the following property, illustrated in Fig. \[Fig2\], that we will call property ${\cal N}_j$ for $j=0,\ldots, r-p-1.$ Let us consider an arbitrary configuration of paths leading from the root $[p+1]$ to some set of vertices at depth $j$. Let us call this part of the tree ${\cal T}_j$, which is now fixed. We pick one designated vertex at depth $j$ (right-most vertex at depth $j$ in Fig. \[Fig2\]). To all other vertices at depth $j$ we attach arbitrary trees $\cal T$ leading to some arbitrary finite sets of leaves in ${\mathbb{N}}^r$ and their multiplicities. We will use the same generic notation $\cal T$ to represent an arbitrary tree, even though they can all be different. The designated vertex has some fixed number of children, say $n_j$, and to each of these children we also attach an arbitrary tree $\cal T$. Property ${\cal N}_j$ will be the following statement.
1. Fix any ${\cal T}_j$ and the number of children $n_j$ of a designated vertex. Suppose that all trees that we just described are good (this means for all choices of trees $\cal T$, possibly empty). Then any tree obtained by adding a single new path leading from a designated vertex at depth $j$ to some new vertex $\alpha\in {\mathbb{N}}^r$ with multiplicity one (as in Fig. \[Fig2\]) is also good.
![\[Fig2\] Illustrating property ${\cal N}_j$. Solid lines represent the subtree ${\cal T}_j$ and $n_j$ children of the designated vertex at depth $j$. Each $\cal T$ represents an arbitrary tree leading to some set of leaves with their multiplicities. Dashed line represents a new path from the designated vertex to a new vertex $\alpha\in {\mathbb{N}}^r$, which has multiplicity one.](figure2.eps){width="40.00000%"}
For any $j=0,\ldots, r-p-1,$ any ${\cal T}_j$ and $n_j$, the property ${\cal N}_j$ holds.
**Proof.** Let us fix any particular choice of trees $\cal T$ attached to non-designated vertices at depth $j$ and $n_j$ children of the designated vertex. By the assumption in property ${\cal N}_j$, this tree, as well as any tree obtained by removing a finite number of leaves, is good. This precisely means that the assumption (M) holds for this tree, or for the sets of leaves with their multiplicities encoded by this tree. Let $Q$ be the matrix described above Theorem \[Sec8Th\] corresponding to this set of leaves. Then, Theorem \[Sec8Th\] implies that $$\sum_{Q'\in{{\cal E}}(Q)} n_j(Q') {\mathbb{P}}(Q^{k+1} = Q')\, M(Q')
\label{Sec9Lem1eq}$$ does not depend on $\omega_{[p+1]}$. Obviously, each $Q'\in {{\cal E}}(Q)$ corresponds to a new tree constructed by adding one more new vertex $\alpha$ to our tree or increasing the multiplicity of some old vertex by one. Moreover, if $n_j(Q') \not = 0$ then the overlap $\alpha \wedge \alpha_\ell$ with one of the old vertices $\alpha_\ell$ should be greater or equal than $j$. This means that this new vertex will be attached to the tree somewhere at depth $j$ or below. One of the possibilities is described in Fig. \[Fig2\], when $\alpha$ is attached by a new path to the designated vertex at depth $j$. All other possibilities – attaching $\alpha$ below one of the non-designated vertices at depth $j$ or below one of the $n_j$ children of the designated vertex – would simply modify one of the trees $\cal T$ in Fig. \[Fig2\]. But such a modification results in a good tree, by the assumption in property ${\cal N}_j$. Since the sum in (\[Sec9Lem1eq\]) is a linear combination of all these possibilities, the term corresponding to adding a new path as in Fig. \[Fig2\] must be good, which finishes the proof.
Next we will prove another property that we will denote ${{\cal P}}_j$ for $j=0,\ldots,r-p-1$, described in Fig. \[Fig3\]. As in Fig. \[Fig2\], we consider an arbitrary configuration ${\cal T}_j$ of paths leading from the root $[p+1]$ to some set of vertices at depth $j$ and we pick one designated vertex among them. To all other vertices at depth $j$ we attach arbitrary trees $\cal T$, while to the designated vertex we attach a single path leading to some vertex $\alpha\in{\mathbb{N}}^r$ with multiplicity one. Property ${{\cal P}}_j$ will be the following statement.
1. Suppose that the following holds for any fixed tree ${\cal T}_j$ up to depth $j$. Suppose that any tree as in Fig. \[Fig3\] is good, as well as any tree obtained by removing any finite number of leaves from this tree. Then any tree obtained by replacing a single path below the designated vertex at depth $j$ by an arbitrary tree $\cal T$ is also good.
We will now prove the following.
Property ${{\cal P}}_j$ holds for any $j=0,\ldots,r-p-1$.
**Proof.** First of all, notice that property ${{\cal P}}_{r-p-1}$ follows immediately from property ${\cal N}_{r-p-1}$. In property ${\cal N}_{r-p-1}$, arbitrary trees $\cal T$ below non-designated vertices at depth $r-p-1$ represent their arbitrary multiplicities, the trees $\cal T$ below the children of the designated vertex are empty, and the multiplicity of the designated vertex is $n_j$. Property ${\cal N}_{r-p-1}$ then implies that we can increase this multiplicity by one to $n_j+1.$ Starting from multiplicity one and using this repeatedly, we can make this multiplicity arbitrary. This is exactly the property ${{\cal P}}_{r-p-1}$.
![\[Fig3\] Illustrating property ${\cal P}_j$. Solid lines represent the subtree ${\cal T}_j$ and one path from a designated vertex at depth $j$ to a leaf $\alpha\in {\mathbb{N}}^r$ with multiplicity one. Property ${\cal P}_j$ allows to replace this single path by an arbitrary tree $\cal T$.](figure3.eps){width="35.00000%"}
Next, we are going to show that property ${{\cal P}}_{j+1}$ implies property ${{\cal P}}_j$. The proof of this is illustrated in Fig. \[Fig4\]. Given any tree as in Fig. \[Fig3\], let us denote by $\delta_j$ the designated vertex at depth $j$. Consider the subtree ${\cal T}_{j+1}$ up to depth $j+1.$ It forms the same pattern, with the child of $\delta_j$ playing a role of the designated vertex at depth $j+1.$ Therefore, by property ${{\cal P}}_{j+1}$ we can replace the single path below this vertex by an arbitrary tree $\cal T$. By property ${\cal N}_j$, if we attach another path to $\delta_j$, the resulting new tree is good. Then we can again treat the child of $\delta_j$ along this new path as a designated vertex at depth $j+1$, apply property ${{\cal P}}_{j+1}$ and replace the path below this vertex by an arbitrary tree. If we continue to repeatedly use property ${\cal N}_j$ to attach another path to $\delta_j$ and then use property ${{\cal P}}_{j+1}$ to replace the part of this path below depth $j+1$ by an arbitrary tree, we can create an arbitrary tree below $\delta_j$, and this tree is good by construction. This is precisely property ${{\cal P}}_j$, so the proof is completed by decreasing induction on $j.$
Finally, this implies Theorem \[Sec9Th1\] (and Theorem \[Sec6iTh1\]). As we explained in Section \[Sec2ilabel\], by induction on $p$ this implies Theorem \[Th2\] .
**Proof of Theorem \[Sec9Th1\].** By Lemma \[Sec2iLem1\], the tree consisting of one path from $[p+1]$ (at depth zero) to some vertex $\alpha\in{\mathbb{N}}^{r}$ (at depth $r-p-1$) with multiplicity one is good. Using property ${{\cal P}}_0$ implies that arbitrary finite tree is good, which finishes the proof.
![\[Fig4\] Illustrating proof of property ${\cal P}_j$. First we replace single path in Fig. \[Fig3\] by arbitrary tree below the child of the designated vertex using ${\cal P}_{j+1}$. Then we iteratively add a new path using property ${\cal N}_j$ and then replace this path below depth $j+1$ be arbitrary tree using ${\cal P}_{j+1}$.](figure4.eps){width="40.00000%"}
[999]{}
Aizenman, M., Contucci, P.: On the stability of the quenched state in mean-field spin-glass models. J. Statist. Phys. **92**, no. 5-6, 765–783 (1998)
Aizenman, M., Sims, R., Starr, S.L.: An extended variational principle for the SK spin-glass model. Phys. Rev. B. **68**, 214403 (2003)
Aldous, D.: Representations for partially exchangeable arrays of random variables. J. Multivariate Anal. [11]{}, no. 4, 581–598 (1981)
Arguin, L.-P., Aizenman, M.: On the structure of quasi-stationary competing particles systems. Ann. Probab. **37**, no. 3, 1080–1113 (2009)
Austin, T.: Exchangeable random measures. To appear, Ann. Inst. Henri Poincaré Probab. Stat., arXiv:1302.2116 (2013)
Austin, T., Panchenko, D.: A hierarchical version of the de Finetti and Aldous-Hoover representations. To appear in Probab. Theory Relat. Fields, arXiv:1301.1259 (2013)
Bolthausen, E., Sznitman, A.-S.: On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys. **197**, no. 2, 247–276 (1998)
Franz, S., Leone, M.: Replica bounds for optimization problems and diluted spin systems. J. Statist. Phys. 111, no. 3-4, 535–564 (2003)
Ghirlanda, S., Guerra, F.: General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity. J. Phys. A [31]{}, no. 46, 9149–9155 (1998)
Guerra, F.: Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. [**233**]{}, no. 1, 1–12 (2003)
Hoover, D. N.: Row-column exchangeability and a generalized model for probability. Exchangeability in probability and statistics (Rome, 1981), pp. 281–291, North-Holland, Amsterdam-New York (1982)
Kallenberg, O.: On the representation theorem for exchangeable arrays. [J. Multivariate Anal.]{}, [30]{}, no. 1, 137–154 (1989)
Mézard, M., Parisi, G.: The Bethe lattice spin glass revisited. Eur. Phys. J. B Condens. Matter Phys. [20]{}, no. 2, 217–233 (2001)
Panchenko, D., Talagrand, M.: Bounds for diluted mean-fields spin glass models. Probab. Theory Related Fields 130, no. 3, 319–336 (2004)
Panchenko, D.: The Parisi ultrametricity conjecture. Ann. of Math. (2) [177]{}, no. 1, 383–393 (2013)
Panchenko, D.: The Parisi formula for mixed $p$-spin models. To appear in Ann. of Probab., arXiv:1112.4409 (2011)
Panchenko, D.: The Sherrington-Kirkpatrick Model. Springer Monographs in Mathematics. Springer-Verlag, New York (2013)
Panchenko, D.: Spin glass models from the point of view of spin distributions. Ann. of Probab. [41]{}, no. 3A, 1315–1361 (2013)
Panchenko, D.: Hierarchical exchangeability of pure states in mean field spin glass models. Preprint, arXiv:1307.2207 (2013)
Panchenko, D.: Structure of $1$-RSB asymptotic Gibbs measures in the diluted $p$-spin models. J. Statist. Phys. 155, no. 1, 1–22 (2014)
Parisi, G.: Infinite number of order parameters for spin-glasses. Phys. Rev. Lett. **43**, 1754–1756 (1979)
Parisi, G.: A sequence of approximate solutions to the S-K model for spin glasses. J. Phys. A **13**, L-115 (1980)
Ruelle, D.: A mathematical reformulation of Derrida’s REM and GREM. Comm. Math. Phys. [**108**]{}, no. 2, 225–239 (1987)
Sherrington, D., Kirkpatrick, S.: Solvable model of a spin glass. Phys. Rev. Lett. [**35**]{}, 1792–1796 (1975)
Talagrand, M.: The Parisi formula. Ann. of Math. (2) **163**, no. 1, 221–263 (2006)
Talagrand, M.: Mean-Field Models for Spin Glasses. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics, Vol. 54, 55. Springer-Verlag (2011)
[^1]: Dept. of Mathematics, Texas A&M University, [email protected]. Partially supported by NSF grant.
| |
We have all come across jobsworths at one time or another. You know the type. The kind of person who is such a stickler for the rules that they apply them without reason or logic. Whether it is the uncooperative service assistant, the over-zealous public official or maybe even your boss at work. These people all share one thing in common. They all know the rules (sometimes a bit to well), they just do not understand the purpose of them.
RULES ARE MADE TO BE BROKEN.
Okay, before everybody descends into complete anarchy, allow me to explain. I will use an example from my old workplace (a bank, which has since gone bust). The rule in question, regards the criteria for allowing employees to take holiday. It is as follows:
1) At least one weeks notice must be given in advance of holiday.
2) No more than two team members can be off at any one time.
It sounds straightforward and is clear to see in black and white. The rule was adhered to without question, which led to an overuse of the word "no" (as well as a vast increase in employee sick days). What the management failed to take onboard when so liberally alienating the staff, was the reason for the rule. Primarily, to ensure that the office was always adequately staffed. Due to a fluctuating workload, what could be defined as adequate staffing changed quite drastically from day to day. If everybody is stretched to their limit and struggling to meet their workload, it makes sense not to allow any more people time off on that day. If there is no work and half of the office are sitting twiddling their thumbs, then the rule is surely obsolete. There are times, like this, when something may contradict a rule, but it does not conflict with the REASON for the rule. This is when common sense should take over and rules should be broken (or simply ignored).
WRITING HAS A LOT OF RULES.
During the first draft, an author lets creativity and inspiration guide his or her hand. When it comes to editing however, they have certain boundaries. This is the time when they have to apply the rules. It is also a process that so many writers and editors get wrong. The problem is that they look solely at the rule and not the PURPOSE of the rule. Basically, the rules of grammar serve three functions. I will call them the three C's.
COHESION, COMPREHENSION and CONSISTENCY.
Grammar and punctuation are really just signifiers. They tell the reader how to read a particular text. Authors should not be consulting text books to decide the placement of a comma, they should be looking inwards, using instinct. One has to think about how they want the reader to read their work. Rhythm and flow are integral to producing an engrossing read. Therefore, do not ask whether a sentence is grammatically correct, but notice how it sounds when you read it back. It can be different for everybody and it is really a question of style. Basically, if I need to take an intake of breath during a sentence, then that will be a comma. Sometimes though. It will be a full stop. This really annoys Microsoft Word, because it will furiously start underlining everything with ugly green pixels and tell you that it is a fragment and to consider revising. I have many arguments with Word and I win every one of them. Why? Because I know what sounds best.
I am currently reading a thriller released through a major publishing house and one thing that I have noticed is that grammatically speaking, it is far from correct. A misuse of the Oxford Comma, missing Signifiers and a lot of fragmented sentences are common throughout. Do I care? No - because the story flows and that is what is important. The story has Cohesion, Comprehension and Consistency. It may not strictly follow the rules, but it sits perfectly with the purpose behind the rules.
One major problem that I see with self published novels is the fact that they are a bit too grammatically correct. What I mean by this, is that because all of the focus is placed on meeting rules, the reason behind the rule is often forgotten. A lot of writers use overly long sentences, which are clunky and awkward to read. When this occurs, I often have to reread a segment, because the meaning was lost in the impractical (albeit correct) sentence structure. When writers have this pointed out to them, they often hide behind the excuse of; "well, it is grammatically correct, so it cannot be wrong" as if it is out of their hands. Nothing is out of the writer's hands. Do not let so called correct punctuation disrupt the flow of your story. Not ever.
A reader will abandon a grammatically correct book, which is dull. They will not abandon an engrossing, exciting book, because the punctuation does not strictly adhere to a certain text on the rules of writing.
THE SINGER, NOT THE SONG
When I read a book, I want to hear the writer's voice. It is as much about the storyteller as the story. When we hear somebody speak with a particular accent, we do not refuse to listen until they start using Received Pronunciation. Why then, do we insist upon forcing a particular form of grammar on them when they write? So long as a story is Cohesive, Comprehensible and Consistent, I could not care if it is grammatically correct or not. Writing is an art form and with all art, part of the beauty is that you can see the individual strokes. These are not blemishes, they are adornments. It is about time that we stopped airbrushing them out.
If you found this post interesting, why not sign up to join my blog using one of the tools on the sidebar to the right. You can also check out my two self published novels The Outback and Stealing Asia. Both are available as ebooks and paperbacks. | http://www.davidclarksonwriter.com/2013/08/the-three-cs-or-why-rules-do-not-really.html |
- In 1930, 28-year-old
Werner Heisenberg wrote
an introduction to a book
where he boldly stated
that the Copenhagen
spirit of quantum theory
has directed the entire development
of modern atomic physics.
Soon Heisenberg's term, Copenhagen Spirit
was reframed as the
Copenhagen interpretation.
And according to Wikipedia,
the Copenhagen interpretation
is "one of the oldest
interpretation in quantum mechanics
"and remains one of the
most commonly taught."
But what was the Copenhagen Spirit
and where did it come from?
Aside from Copenhagen, Denmark of course.
This is mostly the story of the friendship
between Warner Heisenberg and Niels Bohr.
Ready, let's go.
♪ Electricity electricity electricity ♪
When 20-year-old Warner
Heisenberg met 36-year-old
Niels Bohr for the first
time on June 14th, 1922.
He was so excited he wrote his parents,
Bohr is the first
scientists who also makes
an impression as a human being.
He's not just a physicist but much more.
With me he's always especially nice
and he has invited me to see
him once again next week.
Heisenberg then quickly
finished his PhD in Munich
and then got a job as an
assistant to the physicist
Max Born in Goettingen.
Sorry Born and Bohr sound so similar.
At the time Born was working
to try to "find the weak
"points and contradictions"
in Bohr's semi-classical
theory of atoms.
By the end of 1923, Heisenberg
has formed a new theory
and sent it his his friend Wolfgang Pauli
to send it to Niels Bohr.
Bohr immediately responded with
an invitation to Heisenberg
to spend Spring Break
with him in Copenhagen.
Wolfgang Pauli was particularly
pleased to hear this
because he found Heisenberg's theory ugly
and without deep connections.
And Pauli wrote Bohr, I
was therefore very pleased
that you had invited him to Copenhagen.
Hopefully then Heisenberg will return home
with a philosophical
orientation to his thinking.
In Copenhagen, Heisenberg
and Bohr got along like
gangbusters, discussing
philosophy and physics
til midnight every night.
Soon Bohr worked out a
deal so that Heisenberg
could complete his grant
with Max Born in pieces
and then Heisenberg spent many months
of 1924-1925 in Copenhagen.
When Heisenberg returned to Germany
he used some of the ideas
Born in his conversations
in Copenhagen to come up
with a new mathematical
model for quantum mechanics,
which he gave to his advisor Max Born.
Born then realized this was a
math concept called a matrix
and for six months, Heisenberg, Born
and Born's new student,
Pascual Jordan created
a series of papers that
are widely considered
the beginning of modern quantum mechanics.
Although the mathematics
were terribly difficult,
Bohr was as excited as anyone.
Although, according to Heisenberg,
he wasn't very good at math.
Bohr was not a mathematically minded man.
He was, I would say, a
Faraday but not a Maxwell.
As a historian I like that analogy.
Then in January 1926, Erwin Schrodinger
published a wave theory
of quantum mechanics.
And soon there was a fierce
debate between the two methods.
Which everyone assumed were incompatible.
However, by May both
Schrodinger and Wolfgang Pauli
proved that they were
mathematically equivalent.
This gave the advantage to Schrodinger
as his math was much easier
and more importantly,
people, including Schrodinger himself
held the philosophy that
using Schrodinger's theories
quantum mechanics could
be made into a branch
of classical mechanics.
In July, Heisenberg went
to a talk by Schrodinger
and was told to be quiet
as "We are now finished
"with all that nonsense
about quantum jumps."
Depressed, Heisenberg
wrote Bohr that very night
and Bohr then invited
Schrodinger to Denmark
to debate the issue.
Once in Copenhagen even
Heisenberg was surprised
by the ferocity of Bohr's
feelings on the issue.
Bohr now appeared to me
almost as an unrelenting
fanatic, who has not prepared
to make a single concession.
It will hardly be possibly
to convey the intensity
of passion with which the
discussions were conducted
on both sides.
It got so intense that
when Schrodinger fell ill,
Niels Bohr's wife Margarethe
while Niels Bohr continued
the discussions bedside.
Interestingly, Schrodinger
had a far different view
of the encounter.
And a few weeks later we wrote a friend,
"In spite of all out
theoretical points of dispute,
"the relationship with Bohr
and especially Heisenberg
"was totally cloudlessly
amiable and cordial."
Schrodinger was also quite
surprised and a little
bit disappointed to find
that someone of Bohr's fame.
He described Bohr as being
honored almost like a demigod,
would be so shy and timid.
"Bohr talks often for minutes,
almost in a dreamlike,
"visionary and really
quite unclear manner.
"Partially because he is
so full of consideration
"and constantly hesitates
fearing that the other
"might take a statement
of his point of view
"as an insufficient
appreciation of the other."
Despite all the talk,
Schrodinger remained unconvinced.
Then in October 1926, Wolfgang
Pauli wrote Heisenberg
about a strange phenomena.
If he used Heisenberg matrices
then the more he studied
the position of an electron,
then noted with the letter Q,
but now use the letter X.
Then the harder it was to
describe the object's momentum
or mass x speed.
Then and now denoted with the letter P.
Wolfgang Pauli wrote Heisenberg,
"One can look at the world
with the P eye and one can look
"at it with a Q eye.
"But when one would
like to open both eyes,
"then one gets dizzy."
In the beginning of 1927,
while Bohr was on a ski trip
to Norway, Heisenberg
realized that he could derive
this relationship, which he
called the Imprecision Relation.
By mid-March Bohr returned to Copenhagen
and immediately noted problems
with some of the concepts,
which left Heisenberg in tears.
After 10 days, that were
described as very disagreeable,
Heisenberg edited the paper
to Bohr's satisfaction
and published it on March 22nd, 1927.
Bohr happily then sent copies
of the paper to Einstein
with a comment that he felt that it,
"represents most significant
exceptionally brilliant
"contribution to the
discussion of general problems
"of quantum theory."
Einstein was not impressed.
In fact, he hated it with a
passion and Einstein debated
it with Bohr for the rest of his life.
Meanwhile, in about April
1927, Heisenberg and Bohr
were having conflicting arguments
about quantum mechanics.
Heisenberg recalled, "we
are unable to find the same
"language for the
interpretation of the theory."
Part of it was a question of mathematics,
Heisenberg felt that we must realize
that our words don't fit.
They don't really get a
hold in the physical reality
and therefore a new mathematical scheme
is just as good as anything.
Whereas, Bohr was
determined to make it work
outside of mathematical.
His biographer wrote
that Bohr believe that,
"Our words have to fit,
we have nothing else."
Bohr's assistant recalled
that these disagreements
about Heisenberg's work
became a source of inspiration
and Bohr decided that quantum mechanics
needed new works to describe physics
that didn't originate
in classical theories.
According to Bohr's
assistant, Bohr dictated
and the next day all that he
had dictated was discarded
and we began anew.
So it went all summer.
Buy July 1927, Bohr published an article
originally titled The
Philosophical Foundations
of Quantum Theory.
Ironically, Bohr's devotion of the science
of how to communicate quantum
mechanics without math
was incredibly difficult
to follow especially
in the beginning.
So I'm gonna give you his
biographer's paraphrasing of it.
The question of whether
an electron is a particle
or wave is a sensible question
in the classical context.
Where the relations
between object of study
and detector needs no specification.
In Quantum mechanics that
question is meaningless however.
There one should rather ask,
does the electron or any other
object behave like a particle
or like a wave?
That question is answerable
but only if one specificies
the experimental arrangement by means
of which one looks at the electron.
Bohr called this complementarity.
Pairs of complementary
properties which cannot
all be observed or
measured simultaneously.
Like position and momentum
or wave versus particle properties.
The debates about the
uncertainty principle
and Bohr's philosophies got
so intense that the Solvay
Congress decided they
could no longer continue
their embargo of German
scientist and dedicated
their entire meeting to the debate.
At the conference Einstein
would wake up every
morning to give Bohr a
new thought experiment
that was supposed to disprove uncertainty
and Bohr would find a
solution by dinner time.
Meanwhile, by June 1927,
25-year-old Heisenberg
was offered a position as a
full professor in Leipzig,
making him the youngest
full professor in Germany.
Heisenberg became dedicated
to starting a world-class
physics program in Leipzig
with deep connections
to Bohr in Copenhagen
and Born in Gottingen.
And for a few years every
physicist worth his or her salt
learned German and went to
Leipzig, Copenhagen or Gottingen.
And usually all three.
By 1928 any disagreements
between Bohr and Heisenberg
were completely passed.
Heisenberg wrote Bohr
apologizing for his ungrateful
behavior and Bohr replied,
"Rarely have I felt myself
"in more sincere harmony
with any other human being."
In February 1932, the English
scientist James Chadwick
discovered the neutron,
which is a heavy chargeless
material found in the nucleus.
When Bohr found out about it,
he immediately wrote Heisenberg.
Here we have become very
interested in the neutron problem.
Where upon Heisenberg
also wrote his own paper
on the neutron to "apply quantum
mechanics to the nucleus."
Bohr's liquid drop model
ended up having major
implications for the
development of quantum mechanics
and the development of the nuclear bomb.
Heisenberg's theory from this
time, as far as I can tell,
was discounted pretty quickly
as he assumed the neutron
was just an electron and proton
sort of smushed together,
which violated Heisenberg's
own uncertainty principle.
In January 1933, Heisenberg
and Bohr's relationship
got much more challenging
because Hitler came to power.
Heisenberg was convinced
he was a flash in the pan
and tried to convince all
the scientist he could find
to stay in Germany or if
they were forced to flee
that it was only a temporary
thing and maybe they should
just take a leave of absence.
Bohr had no such ideas, was
very concerned about Hitler.
And formed committees,
which eventually helped
over 300 scientists.
Years later Bohr's wife
Margarethe recalled.
Oh the 30s were such terrible years.
There we nearly collapsed.
I remember when we went to
America in '33 we had such
a long list of people for whom
we should try to find places.
In addition, despite the
fact that Bohr wasn't at all
religious, he had something
personal to fear from the Nazis
because his mother was Jewish.
Despite their differences
on politics, Bohr
and Heisenberg continued to
be very close and visit often.
Meanwhile, Heisenberg formed
a very very close relationship
with his student, Carl
Friedrich von Weizsäcker
and according to Heisenberg,
"I saw Weizsäcker almost daily
during the years 1931 to 35."
For example, in October
1934, Heisenberg wrote his
mother that the only way he
could understand the rise
of Hitler was with the
friendship of Carl Friedrich
who struggles in his own serious way
with the world around us.
Heisenberg's relationship
with Weizsäcker got a little
bit more complicated in
1936 when 34-year-old
Heisenberg fell in love with
Weizsäcker's 19-year-old
sister, Adelaide and her
parents rejected the suit.
A few months later Weizsäcker
ended his assistantship
with Heisenberg and started an internship
with Lise Meitner in Berlin.
Their friendship remained
a little distant until 1939
at the start of the war,
when Weizsäcker invited
Heisenberg to join the
Uranium Club to study
the military aspects of nuclear fission.
Back in 1936, Heisenberg
felt very lonely without
the young Weizsäcker.
But then in January 1937,
he met a young woman named
Elizabeth Li Schumaker
and two weeks later they
were engaged.
Heisenberg wrote Bohr
that we would be worried
about being a successful
scientist and family man
if it weren't for the great
example of his idol, Bohr.
After their quick marriage,
Heisenberg brought his new
bride to Copenhagen to meet Bohr.
In Copenhagen Margarethe
Bohr was a bit disturbed
by the politics of her
husband's protege and his wife.
Margarethe took Li on a private
walk away from prying eyes
where there was no fear of being overheard
and said, "Wouldn't it be
nice for you to get rid
"of this terrible government
and get some decent people?"
To which Li replied,
"Oh, but what would we
do without a Fuhrer?"
Margarethe recalled, that
was the attitude of nice
young Germans so I gave up.
As far as I can tell, Niels
Bohr and Werner Heisenberg
did not see each other
for over four years.
But the drought ended in September 1941,
when Werner Heisenberg
went back to Copenhagen
and Heisenberg wrote his wife,
"I am once again in the city
"where part of my heart has stayed stuck."
But much had changed, Germany
had been at war for two years
by this time and although
Denmark had tried
to stay neutral they had
been occupied by the Germans
16 months earlier.
So Heisenberg wasn't just a visitor,
he was a representative
of a repressive regime
that was oppressing their country.
This time Heisenberg's pro-Nazi
comments were too prevalent
to be ignored by Niels Bohr.
And even stranger Heisenberg
told Bohr outright,
that he was working on
nuclear bombs for Hitler.
Which brings up the question, why?
And that mystery is solved next time
on the Lightning Tamers.
Thanks for watching my video,
please remember to give
it a thumbs up and share
it on social media.
And if you really want to be nice
you can become one of my patrons.
Thank you patrons,
there's a link down below.
Okay, remember to stay safe
and be nice to each other.
Have a good day.
(upbeat music)
Which would turned. (rolls tongue)
Oppressed them about 16
months before as well.
(mouth jarbles)
Let me do it again.
That's good.
| |
The entire disclosure of Japanese Patent Application No. 2011-1732, filed January 7 is expressly incorporated by reference herein.
1. Technical Field
The present invention relates to an input selection device which selects an input source to be displayed from a plurality of input sources, a video/sound (video/audio) reproduction system including the input selection device, and a video/sound reproduction method in the video/sound reproduction system.
2. Related Art
A video display device, such as a projector, includes a number of video input terminals or sound input terminals, such that various external video/sound supply devices (input sources) are connectable thereto. A user operates an operation panel in the main body of the device or a remote controller to select a display-target input source from among a plurality of input sources (for example, see JP-A-2003-274316).
When this video display device is used in a place, such as a classroom of a school, there is a case where the video display device is attached at a high position, and sound is output from a comparatively large external speaker. On the other hand, if the video display device is at a high position, it is difficult to connect an input source again in each case, or to select an input source to be displayed. For this reason, an input selection device to which a plurality of input sources are connectable and which can select an input source to be displayed on a video display device is disposed at a position where an input source is easily connected, and video signals and sound signals are respectively supplied to the video display device and an external speaker through the input selection device.
However, if a video input terminal (for example, an HDMI terminal) to which both video signals and sound signals are input are provided in the input selection device configured as above, there is a need for separating the video signals which are supplied to the video display device and the sound signals which are supplied to the external speaker in the input selection device. For this reason, the configuration of the input selection device is complicated, causing an increase in cost.
An advantage of some aspects of the invention is to solve at least a part of the problems described above, and the invention can be implemented as the following forms or application examples.
This application example of the invention is directed to an input selection device including: a plurality of video input terminals to which video signals are input, a plurality of sound input terminals which are associated with the plurality of video input terminals, and to which sound signals corresponding to the video signals are input, an input operation unit which receives a selection operation for selecting one of the plurality of video input terminals, and a control unit which performs control such that video based on a video signal input to a video input terminal selected by the selection operation is displayed on a video display device, and sound based on a sound signal input to a sound input terminal corresponding to the selected video input terminal is output from a sound output device. The plurality of video input terminals include a first video input terminal to which both a video signal and a sound signal are input, and the plurality of sound input terminals include a first sound input terminal which is associated with the first video input terminal.
With this input selection device, the control unit performs control such that video based on a video signal to be input to the video input terminal selected by the selection operation is displayed on the video display device, and sound based on a sound signal to be input to the sound input terminal corresponding to the selected video input terminal is output from the sound output device. For this reason, when the first video input terminal to which both a video signal and a sound signal are input is selected by the selection operation, the control unit performs control such that video based on the video signal to be input to the first video input terminal is displayed on the video display device, and sound based on the sound signal to be input to the first sound input terminal corresponding to the first video input terminal is output from the sound output device. If the video signal and the sound signal which are input to the first video input terminal are output to the video display device as they are, such that video based on the video signal is displayed on the video display device, and the sound signal is separated in the video display device and input to the first sound input terminal, sound based on the sound signal can be output from the sound output device. That is, since there is no need for separating the video signal and the sound signal, which are input to the first video input terminal, in the input selection device when video display and sound output are performed in different devices, it becomes possible to simplify the configuration and to suppress an increase in cost.
It is preferred that the input selection device further includes a plurality of video output terminals which output the video signals input to the plurality of video input terminals to the video display device, and a control signal output unit which outputs a control signal to the video display device. The control unit outputs a control signal according to the selection result of the selection operation from the control signal output unit to perform control such that video based on a video signal input to a video input terminal selected by the selection operation from among the video signals output from the plurality of video output terminals is displayed on the video display device.
It is preferred that the input selection device further includes a sound output terminal which outputs the sound signal to the sound output device, and a sound signal selection unit which determines a sound signal to be output from the sound output terminal from among the sound signals input to the plurality of sound input terminals in accordance with the selection result of the selection operation.
In the input selection device, the first video input terminal may be an HDMI terminal.
In the input selection device, the first video input terminal may be a USB terminal.
This application example of the invention is directed to a video/sound reproduction system including: an input selection device to which video signals and sound signals are input, a video display device which displays video based on a video signal input to the input selection device, and a sound output device which outputs sound based on a sound signal input to the input selection device. The input selection device includes a plurality of video input terminals to which video signals are input, a plurality of sound input terminals which are associated with the plurality of video input terminals, and to which the sound signals corresponding to the video signals are input, an input operation unit which receives a selection operation for selecting one of the plurality of video input terminals, and a control unit which performs control such that video based on a video signal input to the video input terminal selected by the selection operation is displayed on the video display device, and sound based on a sound signal input to a sound input terminal corresponding to the video input terminal selected by the selection operation is output from the sound output device. The plurality of video input terminals include a first video input terminal to which both the video signal and the sound signal are input, and the plurality of sound input terminals include a first sound input terminal which is associated with the first video input terminal. The video display device includes a second video input terminal to which the video signal and the sound signal input to the first video input terminal of the input selection device are input from the input selection device, a separation unit which separates the video signal and the sound signal input to the second video input terminal from each other, a display unit which, when the first video input terminal is selected by the selection operation, displays video based on the video signal separated by the separation unit, and a sound output terminal which, when the first video input terminal is selected by the selection operation, outputs the sound signal separated by the separation unit to the first sound input terminal of the input selection device.
With this video/sound reproduction system, the control unit of the input selection device performs control such that video based on a video signal to be input to the video input terminal selected by the selection operation is displayed on the video display device, and sound based on a sound signal to be input to the sound input terminal corresponding to the selected video input terminal is output from the sound output device. For this reason, when the first video input terminal to which both a video signal and a sound signal are input is selected by the selection operation, the control unit of the input selection device performs control such that video based on the video signal to be input to the first video input terminal is displayed on the video display device, and sound based on the sound signal to be input to the first sound input terminal corresponding to the first video input terminal is output from the sound output device. The video signal and the sound signal which are input to the first video input terminal are input to the second video input terminal of the video display device, and are separated into the video signal and the sound signal by the separation unit of the video display device. The video display device displays video based on the separated video signal by the display unit, and outputs the separated sound signal from the sound output terminal to the first sound input terminal of the input selection device. As a result, sound based on the sound signal is output from the sound output device. That is, since there is no need for separating the video signal and the sound signal, which are input to the first video input terminal, in the input selection device when video display and sound output are performed in different devices, it becomes possible to simplify the configuration of the input selection device and to suppress an increase in cost.
It is preferred that, in the video/sound reproduction system, the input selection device further includes a plurality of video output terminals which output the video signals input to the plurality of video input terminals to the video display device, and a control signal output unit which outputs a control signal to the video display device. The control unit outputs a control signal according to the selection result of the selection operation from the control signal output unit to perform control such that video based on a video signal input to a video input terminal selected by the selection operation from among the video signals output from the plurality of video output terminals is displayed on the video display device.
It is preferred that, in the video/sound reproduction system, the input selection device further includes a sound output terminal which outputs the sound signal to the sound output device, and a sound signal selection unit which determines a sound signal to be output from the sound output terminal from among the sound signals input to the plurality of sound input terminals on the basis of the selection operation.
In the video/sound reproduction system, the first video input terminal may be an HDMI terminal.
In the video/sound reproduction system, the first video input terminal may be a USB terminal.
This application example of the invention is directed to a video/sound reproduction method in a video/sound reproduction system. The video/sound reproduction system includes an input selection device to which video signals and sound signals are input, a video display device which displays video based on a video signal input to the input selection device, and a sound output device which outputs sound based on a sound signal input to the input selection device. The video/sound reproduction method includes causing an input selection device to receive a selection operation for selecting a first video input terminal, to which both the video signal and the sound signal are input, from among a plurality of video input terminals, causing the input selection device to output the video signal and the sound signal input to the first video input terminal to the video display device, causing the video display device to separate the video signal and the sound signal from each other, causing the video display device to display video based on the separated video signal, causing the video display device to output the separated sound signal to the input selection device, causing the input selection device to output the sound signal input from the video display device to the sound output device, and causing the sound output device to output sound based on the sound signal input from the input selection device.
Hereinafter, a video/sound reproduction system of this embodiment will be described with reference to the drawings.
FIG. 1
is a perspective view showing the schematic configuration of the video/sound reproduction system of this embodiment.
FIG. 1
100
1
2
200
3
200
3
3
1
2
1
1
3
2
3
3
1
200
1
100
As shown in , a video/sound reproduction system includes a projector serving as a video display device, a speaker serving as a sound output device, and a switcher serving as an input selection device. An external video/sound supply device (for example, a computer) is appropriately connected to the switcher , and video signals and sound signals are supplied from the video/sound supply device to the switcher . The switcher outputs the supplied video signals to the projector , and also outputs the supplied sound signals to the speaker . The projector of this embodiment is a close projection-type projector using a short focus lens, and is fixed at the upper part of the wall surface. The projector projects (displays) video based on a video signal to be input from the switcher toward a screen SC disposed on the lower side along the wall surface thereof. The speaker is, for example, an active-type speaker equipped with an amplifier, and outputs sound based on a sound signal input from the switcher . The switcher is disposed at a lower position than the projector such that the video/sound supply device is easily connected or disconnected. The projector may be a general-purpose projector which is usable as a stand-alone type or may be a projector exclusively for the video/sound reproduction system .
FIG. 2
1
is a block diagram showing the schematic configuration of the projector .
FIG. 2
1
10
11
12
13
14
15
16
17
18
19
20
21
22
As shown in , the projector includes a video projection unit , a control unit , an input operation unit , a control signal input unit , a video signal input unit , a video/sound separation unit , a sound signal input unit , a video signal selection unit , a sound signal selection unit , a video signal processing unit , a sound signal processing unit , a speaker , and a sound signal output unit .
10
10
10
The video projection unit includes alight source device, a light modulation device such as a liquid crystal light valve, a projection lens, and the like (all of which are not shown). The video projection unit corresponds to a display unit. The video projection unit modulates light emitted from the light source device by the light modulation device to form an image (video), and projects the image from the projection lens on a magnified scale to display the image on the projection surface, such as the screen SC.
11
1
The control unit includes a CPU (Central Processing Unit), a RAM (Random Access Memory) which is used to temporarily store various kinds of data, a nonvolatile ROM (Read Only Memory), and the like. The CPU operates in accordance with a control program stored in the ROM to perform overall control of the operation of the projector .
12
1
12
12
12
11
The input operation unit receives an input operation of the user, and includes a plurality of operation keys which are used when the user issues various instructions to the projector . The operation keys provided in the input operation unit of this embodiment include a power key which is used to turn on and off the power supply, an input source selection key which is used to select an input source, and the like. If the user operates various operation keys of the input operation unit , the input operation unit receives the input operations and outputs operation signals according to the operation contents of the user to the control unit .
13
13
11
11
1
3
13
3
13
The control signal input unit includes an RS-232C terminal, is connected to an external control device, such as a computer, and receives a control signal from the control device as input. The control signal input unit outputs the input control signal to the control unit , and the control unit can control the operation of the projector on the basis of the control signal. The switcher is connected to the control signal input unit of this embodiment, and the control signal is input from the switcher to the control signal input unit .
14
1
2
3
4
5
6
The video signal input unit includes a plurality of video input terminals. In this embodiment, there are two VGA terminals PV and PV to which an analog RGB signal is input from a computer or the like, an S terminal PV to which an S video signal is input from a video apparatus or the like, a composite terminal PV to which a composite signal is input from a video apparatus or the like, an HDMI terminal PV to which a digital video signal or the like based on the HDMI (High Definition Multimedia Interface) standard is input from a video apparatus, a computer, or the like, and a USB terminal (USB Type B terminal) PV to which a digital video signal or the like based on the USB (Universal Serial Bus) standard is input from a computer or the like in which specific software is installed.
5
6
5
6
15
An input signal which is input to the HDMI terminal PV and the USB terminal PV from among the video input terminals may include both a video signal and a sound signal. For this reason, the input signal which is input to the HDMI terminal PV and the USB terminal PV is separated into a video signal and a sound signal by the video/sound separation unit .
16
1
2
3
1
1
1
1
2
2
2
2
3
3
4
3
4
3
The sound signal input unit includes a plurality of sound input terminals. In this embodiment, a plurality of sound input terminals include two mini jacks PA and PA to which an analog sound signal is input from a computer or the like, and a set of pin jacks PA to which an analog sound signal is input from a video apparatus or the like. The sound input terminals are associated with the video input terminals. Specifically, the mini jack PA is associated with the VGA terminal PV, and a sound signal corresponding to a video signal input to the VGA terminal PV is input to the mini jack PA. The mini jack PA is associated with the VGA terminal PV, and a sound signal corresponding to a video signal input to the VGA terminal PV is input to the mini jack PA. The pin jack PA is associated with both the S terminal PV and the composite terminal PV, and a sound signal corresponding to a video signal input to the S terminal PV or the composite terminal PV is input to the pin jack PA.
14
15
17
17
11
19
12
11
17
A video signal input to the video signal input unit and a video signal separated by the video/sound separation unit are input to the video signal selection unit . The video signal selection unit selects one video signal under the control of the control unit , and outputs the video signal to the video signal processing unit . The user operates an input source selection key provided in the input operation unit (selection operation) to select one video input terminal. The control unit instructs the video signal selection unit so as to select a video signal input to the video input terminal selected by the user.
16
15
18
18
11
20
11
18
1
1
2
2
3
4
3
5
6
A sound signal input to the sound signal input unit and a sound signal separated by the video/sound separation unit are input to the sound signal selection unit . The sound signal selection unit selects any one sound signal under the control of the control unit , and outputs the sound signal to the sound signal processing unit . The control unit causes the sound signal selection unit to select a sound signal input to the sound input terminal corresponding to the video input terminal selected by the user. That is, when the VGA terminal PV is selected by the input source selection key, a sound signal input to the mini jack PA is selected, when the VGA terminal PV is selected, a sound signal input to the mini jack PA is selected, and when any one of the S terminal PV and the composite terminal PV is selected, the pin jack PA is selected. When the HDMI terminal PV or the USB terminal PV is selected, a sound signal separated from each input signal is selected.
19
17
10
19
11
10
14
10
The video signal processing unit converts a video signal input from the video signal selection unit to a video signal having a format suitable for forming an image in the light modulation device of the video projection unit . The video signal processing unit performs an image quality adjustment process for adjusting brightness, contrast, or the like on the converted video signal on the basis of an instruction of the control unit as necessary, and outputs the video signal subjected to the process to the video projection unit . As a result, video based on the video signal input to the video signal input unit is projected from the video projection unit .
20
18
11
21
21
22
4
20
1
22
4
20
4
20
21
21
4
21
22
The sound signal processing unit appropriately performs a process, such as a volume adjustment process or a sound quality adjustment process, on a sound signal input from the sound signal selection unit on the basis of an instruction of the control unit . The sound signal subjected to the process is output to the speaker , and sound based on the sound signal is output from the speaker . The sound signal output unit which includes a mini jack PA serving as a sound output terminal is connected to the sound signal processing unit , and can output the sound signal subjected to the process outside the projector . The sound signal output unit includes a detection unit (not shown) which detects whether or not a connection terminal (mini plug) of a sound cable is connected to the mini jack PA, and the sound signal processing unit switches the output destination of a sound signal in accordance with the detection result. That is, when the mini plug is not connected to the mini jack PA, the sound signal processing unit outputs a sound signal to the speaker and causes the speaker to output sound. When the mini plug is connected to the mini jack PA, sound is not output from the speaker , and a sound signal is output to the outside through the sound signal output unit .
FIG. 3
3
is a perspective view showing the switcher .
FIG. 3
3
30
39
30
30
39
30
30
30
1
2
30
30
30
1
30
2
200
30
1
30
2
f
a
t
a
b
b
b
b
As shown in , the switcher is covered by the boxlike casing , and an input operation unit which is used when the user performs an input operation is disposed in a front surface of the casing . The operation surface of the input operation unit is inclined slightly upward, such that the user can easily perform an input operation. A concave portion is formed in an upper surface of the casing , and a plurality of terminals which are used for connection to the projector or the speaker are arranged in the concave portion (the details will be described below). A step is provided in the bottom surface of the casing , and two surfaces (first bottom surface and second bottom surface ) which are different in height are formed. A plurality of terminals which are used for connection to the video/sound supply device are arranged in the first and second bottom surfaces and (the details will be described below).
FIG. 4
FIG. 5
39
3
3
is a front view showing the input operation unit of the switcher . is a block diagram showing the schematic configuration of the switcher .
FIG. 5
3
31
32
33
34
35
36
37
38
39
As shown in , the switcher includes a control unit , a video signal input unit , a video signal output unit , a sound signal input unit , a sound signal selection unit , a sound signal processing unit , a sound signal output unit , a control signal output unit , and an input operation unit .
31
3
The control unit includes a CPU, a RAM, a ROM, and the like, and the CPU operates in accordance with a control program stored in the ROM to perform overall control of the operation of the switcher .
32
14
1
200
32
32
1
2
3
4
5
6
32
30
1
30
FIG. 1
FIG. 3
b
The video signal input unit includes a plurality of video input terminals corresponding to the video input terminals of the video signal input unit of the projector . Video signals are input from the video/sound supply device (see ) to the video signal input unit . That is, the video signal input unit includes two VGA terminals SVa and SVa, an S terminal SVa, a composite terminal SVa, an HDMI terminal SVa, and a USB terminal (USB Type B terminal) SVa. The video input terminals of the video signal input unit are arranged in the first bottom surface (see ) of the casing .
33
32
14
1
32
1
33
1
2
3
4
5
6
32
33
40
33
30
30
14
1
32
3
14
1
t
FIG. 3
The video signal output unit includes a plurality of video output terminals corresponding to the video input terminals of the video signal input unit , that is, a plurality of video output terminals corresponding to the video input terminals of the video signal input unit of the projector , and outputs the video signals input to the video signal input unit to the projector . That is, the video signal output unit includes two VGA terminals SVb and SVb, an S terminal SVb, a composite terminal SVb, an HDMI terminal SVb, and a USB terminal (USB Type A terminal) SVb. The video input terminals of the video signal input unit and the video output terminals of the video signal output unit are correspondingly connected to each other through various circuits , such as a buffer and an amplifier. The video output terminals of the video signal output unit are arranged in the upper surface (see ) of the casing , and are constantly correspondingly connected to the video input terminals of the video signal input unit of the projector through cables (not shown). For this reason, the video signals input to the video signal input unit of the switcher are input to the video signal input unit of the projector in an unchanged format.
34
16
1
200
34
1
2
3
30
2
30
34
4
4
30
30
4
22
1
1
1
1
1
1
2
2
2
2
3
3
4
3
4
3
4
5
6
FIG. 1
FIG. 3
b
t
The sound signal input unit includes a plurality of sound input terminals corresponding to the sound input terminals of the sound signal input unit of the projector , and receives the sound signals from the video/sound supply device (see ) as input. That is, the sound signal input unit includes two mini jacks SAa and SAa, and a set of pin jacks SAa, and these sound input terminals are arranged in the second bottom surface of the casing . The sound signal input unit includes one mini jack SAa, in addition to the sound input terminals. The mini jack SAa is disposed in the upper surface (see ) of the casing , and is constantly connected to the mini jack PA of the sound signal output unit of the projector through a cable (not shown). Similarly to the projector , the sound input terminals are associated with the video input terminals. Specifically, the mini jack SAa is associated with the VGA terminal SVa, and a sound signal corresponding to a video signal input to the VGA terminal SVa is input to the mini jack SAa. The mini jack SAa is associated with the VGA terminal SVa, and a sound signal corresponding to a video signal input to the VGA terminal SVa is input to the mini jack SAa. The pin jack SAa is associated with both the S terminal SVa and the composite terminal SVa, and a sound signal corresponding to a video signal input to the S terminal SVa or the composite terminal SVa is input to the pin jack SAa. The mini jack SAa is associated with both the HDMI terminal SVa and the USB terminal SVa with no corresponding sound input terminal.
34
35
35
31
36
The sound signals input to the sound signal input unit are input to the sound signal selection unit . The sound signal selection unit selects any one sound signal under the control of the control unit , and outputs the sound signal to the sound signal processing unit .
36
35
37
The sound signal processing unit appropriately performs a process, such as a volume adjustment process, on the sound signal input from the sound signal selection unit , and outputs the sound signal subjected to the process to the sound signal output unit .
37
1
36
1
3
1
37
30
30
2
36
2
2
t
FIG. 3
The sound signal output unit includes a mini jack SAb, and outputs the sound signal processed by the sound signal processing unit from the mini jack SAb to the outside the switcher . The mini jack SAb of the sound signal output unit is disposed in the upper surface (see ) of the casing , and is constantly connected to the speaker through a cable (not shown). That is, the sound signal processed by the sound signal processing unit is output to the speaker , and sound based on the sound signal is output from the speaker .
38
30
30
13
1
31
3
1
38
1
t
The control signal output unit includes an RS-232C terminal. The RS-232C terminal is disposed in the upper surface of the casing , and is constantly connected to the control signal input unit of the projector through a cable (not shown). The control unit of the switcher outputs a control signal to the projector through the control signal output unit , and can control the operation of the projector .
39
3
39
1
1
2
3
4
5
39
39
31
4
32
4
1
11
1
2
12
2
13
3
14
4
15
5
16
6
FIG. 4
The input operation unit receives an input operation of the user, and includes a plurality of operation keys or operation knobs which are used when the user issues various instructions to the switcher . As shown in , the operation keys or operation knobs provided in the input operation unit of this embodiment include a power key B which is used to turn on and off the power supply of the projector , a sound mute key B which is used to stop the output of sound, a video/sound mute key B which is used to stop the output of sound and the display of video, a plurality of input source selection keys B which are used to select an input source, and a volume knob B which is used to adjust the volume level of sound. If the user performs various operations on the input operation unit , the input operation unit receives the input operations, and outputs operation signals according to the operation contents of the user to the control unit . The input source selection keys B include a plurality of operation keys corresponding to the video input terminals provided in the video signal input unit , and each operation key is operated to select any one video input terminal. That is, the input source selection key B includes a Computer key B which is used to select the VGA terminal SVa, a Computer key B which is used to select the VGA terminal SVa, an S-Video key B which is used to select the S terminal SVa, a Video key B which is used to select the composite terminal SVa, an HDMI key B which is used to select the HDMI terminal SVa, and a USB key B which is used to select the USB terminal SVa.
3
1
39
Next, the operations of the switcher and the projector when each operation key or operation knob of the input operation unit is operated will be described.
1
39
31
1
38
1
13
12
1
11
1
When the user operates the power key B of the input operation unit , the control unit outputs a control signal (power supply control signal) for turning on and off the power supply of the projector from the control signal output unit to the projector . If the power supply control signal is input through the control signal input unit , similarly to a case where the power key of the input operation unit of the projector is operated, the control unit of the projector turns on and off the power supply.
2
39
31
36
When the user operates the sound mute key B of the input operation unit , the control unit instructs the sound signal processing unit to stop the output of sound.
3
39
31
36
38
1
13
11
1
19
200
11
1
When the user operates the video/sound mute key B of the input operation unit , the control unit causes the sound signal processing unit to stop the output of sound, and also outputs an operation signal (video mute signal) for stopping the display of video from the control signal output unit to the projector . If the video mute signal is input through the control signal input unit , the control unit of the projector instructs the video signal processing unit to switch the display state from a state (normal state) where an input image from the video/sound supply device is displayed to a state (video mute state) where a solid image of a predetermined color (for example, black) is displayed. When the video mute signal is input in the video mute state, the control unit of the projector returns the display state from the video mute state to the normal state.
5
39
31
36
When the user operates the volume knob B of the input operation unit , the control unit instructs the sound signal processing unit to adjust the volume level of sound.
4
39
3
1
When the user operates each input source selection key B of the input operation unit , the switcher and the projector operate as follows in accordance with the operated operation key.
1
11
1
31
1
38
1
13
11
1
17
1
1
31
3
35
1
1
1
1
1
3
10
1
1
2
When the Computer key B is operated and the VGA terminal SVa is selected, the control unit outputs a control signal indicating the selection of the VGA terminal SVa from the control signal output unit to the projector . If the control signal is input through the control signal input unit , the control unit of the projector instructs the video signal selection unit to select a video signal input to the VGA terminal PV corresponding to the selected VGA terminal SVa. The control unit of the switcher instructs the sound signal selection unit to select a sound signal input to the mini jack SAa serving as a sound input terminal corresponding to the selected VGA terminal SVa. As a result, video based on the video signal input to the VGA terminal PV of the projector , that is, the video signal input to the VGA terminal SVa of the switcher is projected from the video projection unit of the projector , and sound based on the sound signal input to the mini jack SAa is output from the speaker .
2
12
13
14
3
1
2
12
2
2
10
1
2
2
13
3
3
10
1
3
2
14
4
4
10
1
3
2
When the Computer key B, the S-Video key B, and the Video key B are operated, the switcher and the projector operate in the same manner as described above. As a result, when the Computer key B is operated and the VGA terminal SVa is selected, video based on the video signal input to the VGA terminal SVa is projected from the video projection unit of the projector , and sound based on the sound signal input to the mini jack SAa is output from the speaker . When the S-Video key B is operated and the S terminal SVa is selected, video based on the video signal input to the S terminal SVa is projected from the video projection unit of the projector , and sound based on the sound signal input to the pin jack SAa is output from the speaker . Similarly, when the Video key B is operated and the composite terminal SVa is selected, video based on the video signal input to the composite terminal SVa is projected from the video projection unit of the projector , and sound based on the sound signal input to the pin jack SAa is output from the speaker .
15
5
31
5
38
1
13
11
1
17
5
5
18
5
4
22
4
34
3
31
3
35
4
5
5
1
5
10
1
4
2
When the HDMI key B is operated, that is, when the HDMI terminal SVa is selected, the control unit outputs a control signal indicating the selection of the HDMI terminal SVa from the control signal output unit to the projector . If the control signal is input through the control signal input unit , the control unit of the projector instructs the video signal selection unit to select a video signal separated from an input signal to the HDMI terminal PV corresponding to the selected HDMI terminal SVa, and also instructs the sound signal selection unit to select a sound signal separated from the input signal to the HDMI terminal PV. The selected sound signal is output from the mini jack PA of the sound signal output unit , and input to the mini jack SAa of the sound signal input unit of the switcher through a cable (not shown). The control unit of the switcher instructs the sound signal selection unit to select a sound signal input to the mini jack SAa serving as a sound input terminal corresponding to the HDMI terminal SVa. As a result, video based on the video signal included in the input signal to the HDMI terminal PV of the projector , that is, the input signal to the HDMI terminal SVa of the switcher is projected from the video projection unit of the projector , and sound based on the sound signal included in the input signal, that is, the sound signal input to the mini jack SAa is output from the speaker .
16
3
1
6
1
6
10
1
4
2
When the USB key B is operated, the switcher and the projector operate in the same manner as described above. In this case, video based on the video signal included in the input signal to the USB terminal PV of the projector , that is, the input signal to the USB terminal SVa of the switcher is projected from the video projection unit of the projector , and sound based on the sound signal included in the input signal, that is, the sound signal input to the mini jack SAa is output from the speaker .
3
200
3
4
3
1
2
The switcher is configured as described above, and the user can connect one or a plurality of video/sound supply devices to the switcher to supply video signals and sound signals. Each connected input source is selected by the corresponding input source selection key B of the switcher , such that video based on a video signal can be displayed on the projector , and sound based on a sound signal can be output from the speaker .
100
As described above, according to the video/sound reproduction system of this embodiment, the following effects can be obtained.
100
31
3
4
1
2
5
6
31
3
5
6
1
4
5
6
2
5
6
5
6
1
15
1
10
4
22
4
34
3
2
5
6
3
3
(1) According to the video/sound reproduction system of this embodiment, the control unit of the switcher performs control such that video based on a video signal input to a video input terminal selected by each input source selection key B is displayed on the projector , and sound based on a sound signal input to a sound input terminal corresponding to the selected video input terminal is output from the speaker . For this reason, when the HDMI terminal SVa or the USB terminal SVa to which both a video signal and a sound signal are input is selected, the control unit of the switcher performs control such that video based on the video signal input to the HDMI terminal SVa or the USB terminal SVa is displayed on the projector , and sound based on the sound signal input to the mini jack SAa corresponding to the HDMI terminal SVa or the USB terminal SVa is output from the speaker . The video signal and the sound signal input to the HDMI terminal SVa and the USB terminal SVa are input to the HDMI terminal PV and the USB terminal PV of the projector , and are separated into the video signal and the sound signal by the video/sound separation unit . The projector projects video based on the separated video signal by the video projection unit , and outputs the separated sound signal from the mini jack PA of the sound signal output unit to the mini jack SAa of the sound signal input unit of the switcher . As a result, sound based on the sound signal is output from the speaker . That is, there is no need for separating the video signal and the sound signal, which are input to the HDMI terminal SVa or the USB terminal SVa, in the switcher when video display and sound output are performed in different devices, it becomes possible to simplify the configuration of the switcher and to suppress an increase in cost.
100
3
30
1
30
2
1
2
30
b
b
t
(2) According to the video/sound reproduction system of this embodiment, in the switcher , a plurality of video input terminals and a plurality of sound input terminals are respectively arranged in two surfaces (first bottom surface and second bottom surface ) which are different in height, such that the user easily recognizes connection positions when connecting the cables, thereby improving convenience. Since the terminals which are connected to the projector and the speaker are disposed in the upper surface , even when the cables are constantly connected, there is no damage to the appearance.
5
6
In this embodiment, the HDMI terminal SVa and the USB terminal SVa serving as a video input terminal, to which both a video signal and a sound signal are input, correspond to a first video input terminal.
The foregoing embodiment may be modified as follows.
5
6
Although in the foregoing embodiment, the HDMI terminal SVa and the USB terminal SVa are used as a video input terminal to which both a video signal and a sound signal are input, the invention is not limited thereto. For example, a network terminal to which a video signal and a sound signal are input through a network, such as a LAN (Local Area Network), may be used. Further, as for the input interface, which both a video signal and a sound signal are input through one cable, such as DP (Display Port) can be used.
14
1
32
3
6
6
33
3
6
200
1
3
200
1
14
1
32
3
33
3
200
1
200
In the foregoing embodiment, the video signal input unit of the projector and the video signal input unit of the switcher include the Type-B USB terminals PV and SVa, and the video signal output unit of the switcher includes the Type-A USB terminal SVb. A video signal and a sound signal output from a USB terminal (Type-A terminal) (not shown) of the video/sound supply device (for example, a computer) are input to the projector through the switcher . In this case, the video/sound supply device functions as a USB host, and the projector functions as a USB device. Meanwhile, the video signal input unit of the projector and the video signal input unit of the switcher may include Type-A USB terminals, the video signal output unit of the switcher may include a Type-B USB terminal, and a video signal and a sound signal may be input from a USB terminal (Type-B terminal) (not shown) of the video/sound supply device (for example, an imaging device or the like). In this case, the projector functions as a USB host, and the video/sound supply device functions as a USB device.
37
3
1
37
1
37
Although in the foregoing embodiment, the sound signal output unit of the switcher includes the mini jack SAb, the sound output terminal provided in the sound signal output unit is not limited to the mini jack SAb. For example, other types of sound output terminals, such as a pin jack, may be provided. The number of sound output terminals provided in the sound signal output unit is not limited to one, and a plurality of sound output terminals may be provided.
34
3
22
1
4
Although in the foregoing embodiment, from among the sound input terminals of the sound signal input unit of the switcher , the sound input terminal to which the sound signal is input from the sound signal output unit of the projector is the mini jack SAa, the invention is not limited thereto. Other types of sound input terminals, such as a pin jack, may be used.
3
1
Although in the foregoing embodiment, the control signal is output from the switcher to the projector through the RS-232C terminal, the invention is not limited thereto. For example, the control signal may be output through the USB terminal.
1
1
Although in the foregoing embodiment, the projector is used as the video display device, the video display device is not limited to the projector . For example, other video display devices, such as a rear projector integrally including a transmissive screen, a liquid crystal display, a plasma display, an SED (Surface-conduction Electron-emitter Display), and an organic EL (Electro Luminescence) display, may be used.
BACKGROUND
SUMMARY
Application Example 1
Application Example 2
Application Example 3
Application Example 4
Application Example 5
Application Example 6
Application Example 7
Application Example 8
Application Example 9
Application Example 10
Application Example 11
DESCRIPTION OF EXEMPLARY EMBODIMENTS
Modifications
BRIEF DESCRIPTION OF THE DRAWINGS
The invention will be described with reference to the accompanying drawings, wherein like numbers reference like elements.
FIG. 1
is a perspective view showing the schematic configuration of a video/sound reproduction system.
FIG. 2
is a block diagram showing the schematic configuration of a projector.
FIG. 3
is a perspective view showing a switcher.
FIG. 4
is a front view showing an input operation unit of a switcher.
FIG. 5
is a block diagram showing the schematic configuration of a switcher. | |
India attempts to reconnect with lunar lander
India’s Chandrayaan-2 mission to the moon launched a spacecraft carrying a rover on July 22, the start of a much-touted mission to explore the unexplored south pole of the moon.
The ending didn’t go nearly as well.
After a long journey, the lander approached the lunar surface on Sept. 7, but scientists from India’s space agency lost contact with it mere moments ahead of the landing.
Now engineers and scientists from the Indian Space and Research Organization (ISRO) are trying to reconnect with the lander to continue the mission.
The attempt to land was going as planned until the Vikram lander was about 2 kilometers above the moon’s surface.
There’s still hope that the lander and Pragyan rover didn’t sustain substantial damage and can continue all or part of the mission.
The latest report from ISRO revealed the lander had a hard landing but remains in one piece. However, it didn’t land flat.
A successful landing would have put India in a elite group of nations that have accomplished a soft landing on the moon, including the United States, the former Soviet Union and China.
“The Chandrayaan-2 mission was a highly complex mission, which represented a significant technological leap,” ISRO said in a statement. “The success criteria was defined for each and every phase of the mission and till date 90 to 95% of the mission objectives have been accomplished and will continue contribute to Lunar science.”
In addition to the lander and rover, the agency also included an orbiting spacecraft in the launch vehicle. The camera on the orbiter has the highest resolution camera (0.3m) in any lunar mission so far and will provide images for the global scientific community.
ISRO says the Chandrayaan-2 mission is “to foster a new age of discovery, increase our understanding of space, stimulate the advancement of technology, promote global alliances, and inspire a future generation of explorers and scientists.”
If space officials aren’t able to reestablish a connection with the lander, this would be the second moon-bound mission of the year that has failed just before landing.
In April, the Israeli Beresheet lunar lander also malfunctioned and failed just before touchdown; that lander was destroyed.
However, this is far from the last attempt to reach the south pole of the moon. NASA is currently planning to send astronauts there in 2024.
There’s a lot of interest in the south pole. Planetary scientists have received new data over the last decade that indicates there are water ice deposits on the south pole. Scientists believe these deposits could be used for life support and to manufacture rocket fuel for future deep-space missions.
The total cost for the Chandrayaan-2 mission has been estimated to be about $145 million. It has been in development for nearly a decade.
India attempts to reconnect with lunar lander
India’s Chandrayaan-2 mission lander was thought to be lost on the surface of the moon, but now ISRO officials are trying to reconnect with it. | https://mothernature.news/2019/09/10/india-attempts-to-reconnect-with-lunar-lander/ |
1. Field of the Invention
The present invention relates to an electrostatic charge image developer for use in developing an electrostatic charge images formed by electrophotography or by electrostatic recording.
2. Description of the Related Art
Various methods for visualizing image information as electrostatic charge images, such as electrophotography, are now widely used in various fields. In electrophotography, an electrostatic latent image is formed on a photoreceptor in two stages, a charging stage and in an exposure stage. The electrostatic latent image is then developed using a developer including a toner and is visualized through transfer and fixation stages. Developers used for this purpose include a two component developer comprising a toner and a carrier and a single component developer used as a single entity, such as a magnetic toner. Of these, the two component developers are widely in use because of advantages such as, for example, that, in the two component developers, the functionalities as the developer are separated as the carrier performs the functions of stirring, transfer, charging, etc. of the developer and that the developer does not include a magnetic powder which is advantageous in color images because the coloring is superior.
In general, toners are manufactured through a mixing and grinding method which includes the steps of molten dispersion of a thermoplastic resin with a pigment, a charge control agent, and a release agent such as wax; cooling; fine grinding; and classifying. In order to improve the flowability and cleanability (the characteristic to be easily cleaned), in some cases, inorganic fine particles or organic fine particles may be added on the surface of the toner particle as necessary.
In recent years, with the rapid development of today's sophisticated information society, there are increasing demands for the ability to provide high image quality information documents constructed in various methods. To address these demands, currently, significant efforts are devoted to researches for improving image quality in various image forming methods. This trend is also true for the image formation through electrophotography, and, in particular, in order to realize a higher resolution image in a color image formation in electrophotography, efforts have been made for data processing techniques of images read by scanners and improvements in digitization techniques in writing with laser. In addition, efforts have been made in research and development of techniques for obtaining smaller size toners, for shaper (narrower) particle size distribution of toners, and for spherical toner particles. For example, when an image is formed using toners having a wide particle size distribution, the toners having a smaller particle size in the particle size distribution cause significant problems such as contamination of developing roller, charging roller, charging blade, photoreceptor, carrier, etc. and spreading of toners. Because of this, it becomes difficult to simultaneously achieve high image quality and high reliability. Such a toner having a wide particle size distribution is also disadvantageous in that reliability is low in a system having functions such as a cleaning function or a toner recycling function. With regard to obtaining a spherical toner particle, the shape of the toner particle significantly affects precision transferability of the toner particle in the transfer stage. That is, the precision transferability becomes higher as the shape is more spherical because the contact area between the toner and the carrier can be maintained at the minimum until the final image is obtained, resulting in a possible improvement in the final image quality characteristics such as reproducibility of fine lines. As described, in order to simultaneously achieve high image quality and high reliability, it is necessary to reduce the particle size of the toner, sharpen the particle size distribution of the toner to obtain a more uniform particle size and a more spherical shape for more uniform surface conditions.
However, because the spherical toners have more uniform surface structures in comparison to randomly-shaped toners such as toners obtained through a grinding process, the distribution of adhering strength with the carrier is narrow, and because the particle size distribution has a narrow width, the distribution of forces applied to the individual toner by a developing electric field has a narrow width. Because of a combination of these two characteristics, as shown in FIG. 1 which shows relationships between the electric filed and amount of development, the spherical toner has a tendency that the developing amount rises, with respect to the developing electric field, in a sharper manner compared to the relationship in randomly shaped toners obtained by grinding. In today's digital color copiers, a latent image is formed by with a laser. In a solid section, the entire surface is exposed and a uniform potential pattern is formed in a wide area. On the other hand, in a halftone region where the image density is lower, the exposure of the laser beam is controlled so that the writing process is performed in a dot-shape or line-shape in order to from a pixel having a very small area of 1 mm2 or less. In addition, the density of the pixel (hereinafter referred to as “input coverage” in the specification) is controlled to reproduce a halftone. The pixels are uniformly exposed with the potential distribution on the photoreceptor and profile of the developing electric field varying depending on the input coverage, that is, in a highlighted section wherein the input coverage is low, the developing electric field is smaller. This does not cause a problem in developing in a region wherein the developing is saturated with respect to the developing electric field such as the solid section and a region of high input coverage. However, in halftone dots having low input coverage, the developing electric field becomes smaller than that for the solid section, and, in some cases, the developing electric field falls out of the saturation range in the electric field-development curve. The electric field felt by the toner depends on the input coverage and becomes lower as the input coverage becomes smaller. Because of this, when a toner having a spherical shape and a narrow distribution in which the density more sensitively responds to the developing electric field is used, in comparison to a ground toner having a wider distribution and random shape, the probability of inability to develop and reproduce in a region of low input coverage is greater. As a result, it is difficult to uniformly develop both the solid section and a region of low input coverage using the spherical toners, and, in some extreme cases, the reproducibility becomes inferior for pixels having an input coverage value smaller than a certain input coverage value. In addition, in some cases, the developing electric field may vary when the distance between the developing roller and the photoreceptor varies due to, for example, deviation of the center. If the developing electric field varies in development of images having an input coverage value of 50 or less on the overall surface, unevenness in images becomes more significant with spherical toners having a sharper development curve with respect to the developing potential compared to ground toners with random shapes. This phenomenon becomes particularly noticeable when the distance between a developing sleeve and photoreceptor (DRS) becomes narrow and the magnitude of the center deviation becomes relatively large compared to the DRS, and also when the peripheral speed of the photoreceptor is high such as in a high-speed copier.
To address this problem, various techniques are employed such as, for example, changing the ratio on the positive side and negative side of an alternating developing bias, but these techniques have not proven too effective so far. Solutions such as an increase in the precision of DRS are not preferable because such solutions causes an increase in cost. Therefore, there is presently no satisfactory method yet.
In order to improve image quality and to significantly reduce toner consumption per page, Japanese Patent Laid-Open Publication No. Hei 11-344837 proposes an electrostatic charge developer comprising a spherical toner having an average volume particle size of approximately 1 μm to 6 μm and a resin-coated carrier having an average volume particle size of 20 μm to 150 μm. In order to obtain high image quality and high image density even when a small size toner is used, Japanese Patent Laid-Open Publication No. 2001-147552 proposes an electrostatic charge developer comprising a toner made by fusion within water-based medium and a resin-coated carrier having a resistivity of 103 Ω·cm–1012 Ω·cm. In this reference, the resistivity of the resin-coated carrier is adjusted by the thickness of the coating layer.
Therefore, an advantage of the present invention is that a developer and an image forming method are provided wherein image quality such as reproducibility of fine lines are improved using spherical toners having a sharp particle size distribution and small particle size while eliminating reproduction deficiencies in low input coverage sections which is a disadvantage of spherical toners and wherein density unevenness can be inhibited even in a document containing entirely halftone images.
| |
This art appraisal report offers an in-depth and impartial assessment of the artwork in question, grounded in the appraiser’s expertise and familiarity with the art market. All the information and data analyzed in this report is sourced solely from the client.
Having a clear understanding of the value of your artwork is crucial in making informed decisions about its future. This report provides a precise estimate of the value of each piece, using US dollars as the base currency. It is not intended to encourage the sale of the artwork, but rather to provide valuable information on how to proceed should the client decide to do so in the future.
Detailed description of the artwork, including its medium, dimensions, and condition.
Checking Originality: Identification with Artificial Intelligence Test
In the quest to identify a match, Image Search employs advanced AI techniques to scour databases of images in order to find visually similar images. This is achieved through the use of various algorithms such as pattern recognition and machine learning. While some results may be considered as “matches” due to a clear similarity, other results may be inconclusive as they rely more on chance rather than any specific similarities. To conduct this test, a front-facing image was used as a reference to search for similar images on the internet.
The results of the automatic recognition are not conclusive. If a match is found, it will be shown below:
What specific information can we obtain from this test?
The algorithm found an exact match. This result is associated with prints, either regular or limited edition prints, that were hand signed by the original artist of this piece of art. It’s important to determine what type you have, so I need to go through the research and inspection process.
Oskar Kokoschka
Poster for the Kunstschau exhibition, Vienna
1908
Printed by Lith. Anst. A. Berger, Wien
Age estimation
In order to ascertain the age of an original Lithograph Poster by Oskar Kokoschka CBE for the Kunstschau exhibition, Vienna 1908, it is necessary to examine various characteristics of the painting. Firstly, its frame construction can be used to accurately date the artwork, as the type of frame used will indicate the era in which it was created. The color palette used is also indicative of the time period in which the painting was created, as certain color combinations and techniques were more popular in certain time periods. Additionally, detail and accuracy of the painting's composition can be used to identify its age, as the level of detail and accuracy of the artwork will reflect the techniques used in that particular era. Finally, the signature of the artist can be used to determine the age of the painting, as the signature style used by Oskar Kokoschka CBE would have changed over time. Therefore, by examining these characteristics, it is possible to determine the age of the painting with a high degree of accuracy.
Based on this information and the pictures provided, I can estimate this painting was made circa 1908.
Condition of the artwork
This Original Lithograph Poster by Oskar Kokoschka CBE is in excellent condition. It was produced for the Kunstschau exhibition in Vienna in 1908 and there is no need for any restoration. There is no appreciable damage and if there is any, it is minimal. This poster is a great example of Kokoschka's art and would be a great addition to any art collection.
Artist’s name, biographical information, artwork’s provenance (history of ownership) and exhibition history.
I study and research the signature of artwork to see if it matches any known signatures. This is an iconic lithograph by well known artist.
Oskar Kokoschka CBE was a renowned Austrian artist, poet, playwright, and teacher of the 20th century. He is best known for his intense expressionistic portraits and landscapes, which often evoke emotion through their vivid colors and bold brushstrokes. He was also an influential theorist on vision, which had a strong effect on the Viennese Expressionist movement. Kokoschka was born on March 1, 1886 and died on February 22, 1980. Throughout his long career, he managed to create a vast body of work that continues to captivate and inspire audiences today.
Detailed analysis of the artwork’s style, subject matter, and significance within the artist’s oeuvre and the broader art world.
I can check if the style and type of painting match those of the artist referenced.
The painting by Oskar Kokoschka CBE for the Kunstschau exhibition in Vienna 1908 is a stunning example of his expressive, bold style. His brushstrokes create dynamic shapes and vibrant colors, creating a distinct sense of motion and emotion. His strong use of light and dark creates a dramatic contrast, further emphasizing his vivid and energetic composition. He expertly combines elements of abstraction and figuration, creating a unique and dynamic visual experience. The painting is a captivating example of his passionate and innovative style.
Comparable sales information, including prices realized at recent auctions or private sales of similar works by the artist or in the same medium.
In order to provide an up-to-date estimate of the fair market value for the original lithograph poster by Oskar Kokoschka CBE for the Kunstschau exhibition, Vienna 1908, I utilized the data collected, including auction prices and other relevant market information. This is crucial as it can be used in various contexts such as insurance, estate planning, and art market analysis. It also offers a valuable insight into how the valuation of the artwork may have changed due to environmental or economic factors such as the current market demand for the work of Oskar Kokoschka CBE.
The auction prices were a significant factor in determining the current market value of the artwork, as they are based on actual transactions between buyers and sellers in the art market. As such, they are a strong indicator of the expected value of the piece in the near future.
By analyzing auction results from the last 6 months, I was able to accurately determine the current fair market value of the artwork. This approach provides a comprehensive view of how the value has changed over time and gives insight into any potential areas of appreciation or depreciation in its price. Additionally, it allows me to adjust my valuation as new auction prices become available.
Conclusion
Investing in art can be a great idea for many reasons. Firstly, investing in art can be a great way to diversify your portfolio, as it is not correlated to the stock market or other investments. Additionally, artwork can be seen as an appreciating asset, meaning it can increase in value over time. Furthermore, collecting art can be a way to support and promote the work of talented artists. Finally, buying a piece of artwork can be a great investment as it is something that can be enjoyed and can provide a great source of pleasure. An original lithograph poster by Oskar Kokoschka CBE for the Kunstschau exhibition in Vienna 1908 is a perfect example of an artwork that could provide both financial and aesthetic value.
This painting can be considered valuable by the art market because it is an original lithograph poster by Oskar Kokoschka CBE, one of the most renowned Expressionist painters of the 20th century. This poster was created for the Kunstschau exhibition in Vienna in 1908, making it a rare and important piece of art history. Furthermore, it is a unique example of Kokoschka's early work, which is highly sought after by collectors. As a result, this painting has great value and is likely to appreciate in value over time.
Final Appraisal Value ($)
12,000-15,000$
Appraisal Report made by:
Andrés Gómez
BSc, MSc, Expert Art Appraiser
10+ years of experience in Online Art Appraisals
100k+ Customers Served
Antique Store Owner
You can check my portofolio of past appraisals here:
https://www.appraisily.com/andres-portofolio/
Relevant photographs or supporting documentation, such as condition reports or expert opinions
A detailed summary of the appraisal process and the appraiser’s qualifications.
Mark-to-market art appraisal is a vital method for determining the current value of a piece of artwork. This form of valuation requires an appraiser to consider various factors, such as market conditions, the condition and age of the artwork, and the artist’s reputation. By taking all these elements into account, a mark-to-market art appraisal delivers an accurate assessment of a piece of artwork’s current market value.
The artist’s reputation, as determined by their track record in gallery and museum shows, awards, and other accomplishments, is also considered in mark-to-market art appraisal. Appraisers use this information to determine if the value of a piece is likely to increase or decrease over time. Additionally, they will inspect the condition of the artwork and note any signs of wear or damage that might affect its future resale value.
When performing mark-to-market art appraisals, appraisers also consider market conditions by researching current art market trends and comparable works that have recently sold. This information is used to provide an estimate of a piece’s worth at that point in time. By considering all of these factors, mark-to-market art appraisal is able to give a reliable indication of the current value of a work. This kind of valuation can also ensure fair prices are paid and received when buying or selling art.
In summary, mark-to-market art appraisal is a crucial tool for determining the true value of a piece of artwork, enabling buyers, sellers, and appraisers to make informed decisions regarding its worth. It takes into account multiple aspects to provide an accurate assessment of the current market value of a work. This information can be used to ensure that buyers and sellers are getting a fair price for the artwork, and that the appraiser’s valuation is up-to-date and reflective of current market conditions.
In the case of insurance replacement appraisals, mark-to-market art appraisals can also be used to accurately estimate the cost of replacing a lost or damaged artwork. The current value, as determined by the appraisal, is then used to determine the amount that the insurance company will pay back to the policyholder. This way, policyholders can rest assured that they will receive an appropriate sum for any artwork that needs to be replaced due to accidental damage or theft. Additionally, this kind of valuation helps insurers ensure they are not being overcharged when artwork needs to be replaced as part of a claim settlement.
The appraisal process is a thorough evaluation of the item or items in question. It involves researching and analyzing the information provided by the requester in order to provide an accurate estimate of its value. The appraiser takes into account factors such as condition, rarity, demand, and market prices. Photographs and detailed descriptions are especially important when providing an appraisal, since they help the appraiser identify any potential flaws or defects that could affect the item’s worth. By using all the resources that are available, an evaluation can be done quickly, efficiently, and with a high level of accuracy.
A statement of the appraiser’s liability and any potential conflicts of interest.
A qualified art appraisal, also known as a formal written evaluation, is a professional assessment of the monetary value of a piece of art by an individual who has specialized knowledge, expertise, and training in the field of art appraisal. This person must meet certain educational and professional requirements, including experience in researching and evaluating art, as well as knowledge of the art market and current market trends. The purpose of a qualified art appraisal is to provide an objective and unbiased opinion of the value of a piece of art for various purposes, including insurance claims, tax planning, estate planning, or to help determine a fair price for a sale or purchase.
We are committed to providing our clients with the most accurate and unbiased appraisal reports. To ensure impartiality, we adopt a flat rate, fixed fee structure for all appraisals, instead of a percentage-based fee. This eliminates any potential conflicts of interest between the art appraiser and the final report value. Our appraisal reports are in compliance with the Appraisal Foundation’s USPAP (Uniform Standards of Professional Appraisal Practice) standards and guidelines, which are widely accepted as the ethical and performance standards for appraisers. This guarantees that our reports are of high quality and legally defensible.
How to sell this artwork.
We have a structured guide to help you sell your artwork, you can find it here. | https://www.appraisily.com/appraisals/an-original-lithograph-poster-by-oskar-kokoschka-cbe-1-march-1886-22-february-1980-for-the-kunstschau-exhibition-vienna-1908/ |
Here is this week’s math problem for you to solve!
This week, we move on to a new topic: exponents.
Use the figure above to solve the three expressions. An exponent is how many times you multiply the base number by itself. Stuck? Need to check your answers? Just click “Continue Reading” below.
Thanks for reading! Come back next week for a new problem!
1. You would multiply the base number 2 three times and get an answer of 8. (2x2x2).
2. You would multiply the base number 5 two times and get an answer of 25. (5×5).
3. You would multiply the base number 7 two times and get an answer of 49. (7×7). | https://12andbeyond.com/2015/07/31/math-problem-of-the-week-august-2/ |
First, it has been established that group dynamics and, in particular, coevolution with group size can favor the evolution of cooperation. However, such work makes assumptions that are not realistic for some well-known cooperative breeders such as African wild dogs (Lycaon pictus) and meerkats (Suricata suricatta). Most notably, it remains unclear how group size and cooperation coevolve in stable groups with complete reproductive skew. Here, I develop and analyze a model of a social group with robust ecology based on social carnivorans. I find that, under a range of conditions, increased cooperation will evolve. Importantly, this result provides a potential mechanism for the evolution of cooperation in groups with only one reproductive individual without invoking kin selection or group selection arguments. Second, territorial breeders are widespread and occur across taxa. Often, these species are characterized by males establishing territories to which they attract females. There exists theory for how territory establishment occurs, but such work ignores the subsequent choice females make in deciding with whom to mate. Here, I develop and analyze a spatially explicit model of male territory establishment and female mate choice. I find that sexually antagonistic selection (i.e., a negative correlation between male and female fitness) occurs in a suite of parameter combinations and model assumptions. Importantly, I find that this is the result of an edge effect on the nite landscape. | https://www.evolbio.mpg.de/events/16722/2169 |
Q:
Rotating a point on a circle
The wheels on a bicycle have $r$-inch radii. After the front wheel picks up a tack, the bike rolls for another $d$ feet and stops. How far above the ground is the tack?
I've been thinking about this problem for a couple of days and I keep coming up against the same issue. I can do the problem fine as long as I have actual numbers for $r$ and $d$, but in the general case I'm having trouble figuring out how to find the angle of rotation. Dividing $d$ by the circumference gives the number of times that the circle rotates, and then to find the angle of rotation I just need multiply the fractional part of this number by 360. In the general case, I keep getting stuck on how to convey the fractional part of a number. I've found that the total number of rotations made is $\dfrac{6d}{r\pi}$; is there a way to notate the fractional part of this number? Is there a way to do this so that I don't need that?
I could write it as $\dfrac{6d}{r\pi}-\lfloor \dfrac{6d}{r\pi}\rfloor$ but I'd like to avoid introducing floor notation if possible.
A:
The number of rotations can be found by $\frac{d}{2\pi r}$, or the distance divided by the circumference of the wheel.
Multiplying by $2\pi$ gets you the total number of radians: $\theta = 2\pi \cdot \frac{d}{2\pi i} = \frac{d}{r}$
The vertical coordinate of the tack with respect to the center of the wheel is generally given by $r \sin \left(\theta \pm \frac{\pi}{2}\right).$ The $\pm \frac{\pi}{2}$ is a phase shift to describe the fact that the tack is on the ground when it gets picked up. The $\pm$ part depends on whether you consider positive angles to go counterclockwise (i.e. bike traveling to the left) or clockwise (i.e. bike traveling to the right)
To get the distance from the ground simply add the wheel's radius.
| |
Honey-roasted sweet potatoes with cinnamon and ginger pair perfectly with our favorite lemon zest labneh recipe, all finished with a generous drizzle of good olive oil and flaky sea salt for a simple, impressive side!
Roasted Sweet Potatoes + Labneh: The Perfect Match
There’s nothing quite like crispy, hot, roasted sweet potatoes fresh out of the oven, except for maybe when they have been tossed in good olive oil, honey and warm spices. We enjoy oven-roasted sweet potatoes year round in our family, and this is my new favorite way to serve them.
A few years ago, Natalie discovered an amazing recipe in the Gjelina cookbook. The recipe pairs labneh (a soft cheese made from strained yogurt) with roasted sweet potato wedges. Soon after, Natalie made the dish for a party I was hosting and we all fell in love with the idea of serving sweet potatoes with labneh. We have been busy perfecting this simple sweet potato recipe ever since, and I am confident you’ll love it as much as we do!
How To Roast Sweet Potatoes:
Roasting vegetables is one of the easiest ways to cook them, and a great way of adding vitamins, fiber, and FLAVOR to your dinner! Sweet potatoes can be roasted whole, in wedges, or in cubes. For this dish, we've opted for baked sweet potato cubes. Here’s the how to roast sweet potatoes in the oven:
- Preheat oven to 425°F.
- Mix honey, olive oil, salt, cinnamon, ginger, and red pepper flakes in a bowl.
- Peel sweet potatoes and cut them into cubes.
- Place the sweet potatoes on a rimmed baking sheet and toss them in the honey mixture until they're coated.
- Bake until the potatoes are tender and slightly caramelized. How long this takes will depend on the size of the potato pieces.
How Long Do You Roast Sweet Potatoes?
How long you roast your sweet potatoes is entirely dependent on the size of the potato pieces you are baking. If you’re baking wedges you’ll need to cook them for 50-60 minutes. If you’re making roasted sweet potato cubes, your baking time is cut in half—roughly 25-30 minutes— which the reason we decided to call for sweet potato cubes in this recipe. The smaller you go, the less time it takes.
A few more favorite variations of sweet potatoes that we’ve cooked up in the oven :
Half-Inch Dice: Like in our favorite Sweet Potato Tacos.
Thinly Sliced: Just like these in this Healthy Sweet Potato Casserole.
What Is Labneh?
I’d like to introduce you to the roasted sweet potatoes' greatest sidekick, homemade labneh cheese. If you’ve already tried this thick, creamy, spreadable yogurt cheese then you know it’s what dreams are made of.
Although labneh (or lebneh) is considered a cheese, it is made simply from plain yogurt that has had some of the moisture strained out of it. It is light and refreshing with a tart and mild flavor. It has been a traditional part of Lebanese cooking for centuries, but has been widely adopted outside of Middle Eastern cuisine, as well.
Eaten much like hummus, labneh is traditionally included as a part of any mezze spread. It is often served as a dip for falafel or pita, a protein-packed breakfast with fruit, or served with olive oil and veggies for lunch. This labneh recipe is extremely versatile. I like to think of it as a healthier substitute for cream cheese or sour cream, with many additional benefits and nutrients.
How To Make Labneh:
Making labneh is so fun, and it's incredibly easy to prepare at home! The only “special” tool you need to make it is a clean kitchen towel. I like to use the thin, flour sack kind of towel, but if you happen to have some layers of cheesecloth or a nut milk bag, either one of these will work as well.
Step 1: start with a tub of plain, whole milk Greek yogurt, some salt and a lemon.
Step 2: Line a colander with the towel, cheesecloth, or nut milk bag and set it over a large bowl.
Step 3: Scoop the yogurt into the lined colander and place it in the fridge to strain. The longer you strain your yogurt (up to 24 hours) the thicker it will get. We like ours soft and spreadable, so for this particular simple sweet potato and labneh recipe, we only strain it for a few hours.
This is Best Sweet Potato Recipe You’ll Ever Make!
Now that you know how to make both labneh and honey-roasted sweet potatoes, it’s time to put it all together! We promise, you are going to love this delicious sweet potato side dish.
Here’s a little recap of everything that we've got going on that makes this the best sweet potato recipe ever:
- Cinnamon, ground ginger, salt, and crushed red pepper flakes, oh my!
- Honey and olive oil, for the win!
- Labneh yogurt, made with lemon zest, salt, and your favorite plain Greek yogurt.
- Flat leaf parsley and green onions to help this healthy oven-roasted sweet potato dish achieve over-the-top greatness!
How Healthy Is This Roasted Sweet Potatoes Recipe?
Not only do these roast sweet potatoes with labneh taste phenomenal, it’s actually really good for you! Take these facts into consideration when planning this dish and you’ll feel great that you found such a winning combo of ‘delicious-meets-good-for-you’ food.
These healthy, oven-roasted sweet potatoes are full of nutrients, such as:
- Lots of vitamin A!
- Vitamin C and beta-carotene (immunity boost anyone?)
- Fiber and potassium!
- Vitamin B6, calcium, iron, magnesium—the list of health benefits goes on and on!
Labneh boasts:
- More protein with fewer calories, fat, and sodium than similar dairy options like cream cheese, sour cream or creme fraiche.
- Good-for-your-gut probiotics (another great immunity booster)!
- Calcium!
Wondering What To Serve With This Roasted Sweet Potato Recipe?
So, now that I’ve convinced you to make these oven-roasted sweet potatoes with labneh cheese, you’re wondering what to serve them with, right? Well, they could be served as a light dinner in and of themselves. But they'd also be delicious served alongside these fantastic main dishes.
- Lamb Burger With Harissa Mayo and Pickled Red Cabbage. You can swap out the goat cheese on the burger for an extra dollop of labneh!
- Beef Kofta Kababs with Tzatziki pair perfectly with this sweet potato side dish.
- Grilled Chicken Shawarma Kebabs- say hello to your new favorite dinner!
In Love with these Honey-Roasted Sweet Potatoes?
Did you make and LOVE this dish? We want to see your labneh and oven roasted sweet potatoes creations! Share your handiwork with us on Instagram and tag @themodernproper and #themodernproper so we can feature you on our stories. Happy eating!
Honey Roasted Sweet Potatoes with Labneh
Serves 6
Ingredients
|16 oz||plain greek yogurt|
|1||lemon, zested|
|2 tsp||salt, divided|
|3 lbs||sweet potatoes, peeled cut into 1 1/2 " pieces|
|3 tbsp||honey, warmed|
|⅓ cup||olive oil|
|1 tsp||cinnamon|
|1 tsp||ground ginger|
|½ tsp||crushed red pepper flakes|
|flat leaf parsley, chopped|
|green onions, chopped|
|olive oil|
|flaky sea salt|
Method
- In a small bowl, mix the yogurt, 1 teaspoon salt and lemon zest.
- Line a large sieve or colander with cheesecloth or clean, thin (flour sack style) dish cloth; set over a large bowl. Place the yogurt mixture in the colander. Gather edges of cheesecloth to cover yogurt. Place in refrigerator and let drain for 4-6 hours (longer if you want it to be thicker).
- Gently squeeze out any excess liquid over the sink, set aside.
- Heat oven to 375 degrees F.
- In a small bowl mix the honey, olive oil, cinnamon, ginger, salt and crushed red pepper flakes.
- Lay the sweet potatoes out on a rimmed baking dish in a single layer. Drizzle the honey mixture over the potatoes. Roast, tossing occasionally, for 35-40 minutes in oven or until tender.Drizzle with more extra-virgin olive oil.
- Spread the labneh onto a platter and top with the roasted sweet potatoes. Drizzle with extra olive oil and sprinkle with parsley, green onions and flakey sea salt to taste. | https://themodernproper.com/posts/honey-roasted-sweet-potatoes-with-labneh |
Can someone PLEASE HELP ME!!! I purchased a new mobile phone and got a new mobile number as well... How do I restore my contacts from my old device to my new device that has a new mobile number as well
I'm most definitely NOT a VZW employee. If a post answered your question, please mark it as the answer.
olds phone had back up assistant on it
old phone was a cosmos
Hi murryrob,
The new phone number would only be linked to your Backup Assistant on the old phone if you changed your number before changing the phone. You can run Backup Assistant on the new phone to see if anything transfers over. If nothing does, your option will be to take both phones to a Verizon store where they can transfer the contacts from one phone to the other.
Thanks, | https://community.verizon.com/t5/Android-Apps/Restoring-my-contacts-with-back-up-assistant/td-p/271258 |
I ended our last blog writing about the cruisers on Big Run, Sharon and Bob, and how we might not see them again. In a large anchorage, they were right next to us in their trawler and we happened to sit down at DeShaMon Restaurant next to them. This morning, Mark took Daisy ashore and instead of going to Government Dock where the dinghies can tie up, he went to a beach near us. Coming back to the boat, he passed Big Run. They stopped him and asked if we wanted to go south with them today. They were leaving in a half hour, and it usually takes us at least 45 minutes to get the dinghy up, roll the sun shades up and get the boat ready to sail. Mark didn’t want to hold them up, so he said we weren’t quite ready and they left a short time later. Our plan was to leave later in the day to go to Big Farmer’s Cay, a several hour sail. However, we decided to get ready to go quickly and try to catch up to Big Run. They were going through some very shallow water in the Pimlico Cays today which we couldn’t do without someone leading us through, since we had never dared take that inside route. Normally we would go outside on the Exuma Sound for that leg of our south run. But they had been through those waters before and had good waypoints. If we could follow them, it would be a new experience and give us an alternative route. Their draft is only one inch less than us so if they weren’t going aground neither would we.
Eventually we caught up to Big Run and they took us through some very skinny water. At one point there was less than a foot under our keel and we were at high tide. The colors of the water are incredible, varying by depth.
We went all the way to Lee Stocking Island, which used to have a research center on it. We took a tour of it three years ago. Now there is no one on the island, but there are still mooring balls there and they were filled with about ten boats.
We ended up at a small anchorage north of Children’s Cay. We left at 0904 and dropped the anchor at 1420. For part of the way we went very slowly as we watched depths.
Our two boats were the only ones anchored in this very quiet, peaceful place. The wind dropped and the water was smooth. Big Run was to the left of us in this photo. It was taken from the hill we climbed to go over to the Exuma Sound side of the cay. The photo above is us at the top of the hill overlooking the Sound. The views from this side of the islands are dramatic with waves crashing up on the rocky shore. This island has some sandy beaches on the Exuma Sound side, so we climbed down and walked on them. Daisy found a big stick to chew on.
After we finished our walk we climbed back to the top of the hill and realized Daisy’s Gentle Leader Harness was not on her. We had taken her leash off on the deserted beach, but left her harness on. Mark walked back and found it at the farthest place we had walked, the very spot where Daisy was chewing on the stick. If you look very closely at the photo I took of her, you’ll see her Gentle Leader just to the left of her head. It probably fell off when she was digging for the stick.
Back on the boat, we invited Sharon and Bob over for grilled burgers. I had made whole wheat hamburger buns while we were on our way south. As you can see, Daisy has found a new friend.
Big Run invited us to join them tomorrow on their trip to Long Island. We were going to invite ourselves to join them anyway, so we quickly accepted. We wanted to go to Long Island when we were in Georgetown in 2010, since it’s a short trip from there, but we ran out of time. It is great to be going with someone who has been there.
Click here for some photos of the beautiful scenery we saw today. | http://seasthedaynow.com/Seas_The_Day/Blog/Entries/2013/2/13_Blackpoint_to_Lee_Stocking.html |
Q:
About $0!=1$ and $a^0=1$ as cases of empty product.
Some useful ''conventions'' as $0!=1$ or $a^0=1$ are particular cases of an empty product, i.e. a product between elements of the empty set. I know that such product is defined as a convention by:
$$
\prod _{x_i\in \emptyset} x_i=1
$$
This convention is usually motivated (see here) by the fact that the definition of the product of $n$ numbers $\{x_1,\cdots x_i,\cdots x_n\}$:
$$
P_n=\prod _{i=1}^n x_i
$$
can be given recursively as $P_n=x_n P_{n-1}$ and the recursive definition become simpler and ''universal'' if we assume $P_0=1$ i.e.: the product with no factors is $1$.
But i don't understand how this convention can be reconciled with the axiomatic definition of the product operation in a field ( or ring) $R$. As far as I know the product is defined as a binary operation $\cdot :R\times R \rightarrow R$ and the empty set is not an element of $R\times R$.
So, my question is: the definition of the empty product is only a notation convention that has not an exact mathematical meaning ( since it can not be derived by the axioms) or it can be given in some way as an axiom when we define a ring? And, in this case what kind of axiom we must introduce so that $0!=1$ and $a^0=1$ became theorems and not simply notational conventions?
A:
It can be instructive to look at the set-theoretic product here. Indeed, when the natural numbers are formalised as sets, it's true that the product of natural numbers is (the isomorphic to) the product of the corresponding sets; and multiplication in a field, commutative ring, algebra or whatever, can be considered to generalise that of natural numbers. So the set-theoretic empty product tells us what the empty product in a ring 'should be'.
Given a set $S$ of sets, the product $\prod S$ is the set of choice functions for $S$, i.e. the set of functions $f : S \to \bigcup S$ such that $f(X) \in X$ for all $X \in S$. In more familiar terms, i.e. if $S = \{ X_i \mid i \in I \}$ for some index set $I$, the product $\prod S = \prod_{i \in I} X_i$ is the set of functions $f : I \to \bigcup_{i \in I} X_i$ such that $f(i) \in X_i$ for all $i \in I$... but for talking about the empty set, the $\prod S$ notation will be more useful.
Now, consider the case when $S = \varnothing$. Then $\bigcup S = \varnothing$ too, so $\prod \varnothing$ is a set of functions $\varnothing \to \varnothing$. There is only one such function, namely $\varnothing$ itself (when functions are formalised as sets of ordered pairs) and it vacuously satisfies the 'choice function' condition; hence $\prod \varnothing = \{ \varnothing \} = 1$.
So considering multiplication in an arbitrary commutative ring (or module or monoid or algebra or ...) as a generalisation of multiplication of natural numbers, in addition to the useful properties mentioned by other users, it makes sense to consider the empty product as being to the multiplicative unit.
This carries over to factorials and exponentials immediately. Indeed:
$n!$ is the number of permutations of a set of size $n$. Thus $0!$ is the number of permutations of the empty set, namely $1$ (the empty function).
$m^n$ is the number of functions $X \to Y$ where $|X|=n$ and $|Y|=m$. When $n=0$, $m^0$ is the number of functions $\varnothing \to Y$, which is... $1$, again! (This definition also yields $0^0=1$.)
| |
Posts tagged with "Flowers"
Tag: Flowers
♥My Beautiful Lilacs,Bleeding Hearts and Other Spring Flowers in the Early...
Apr 1, 2015
Gardening Tips & Flowers : How to Small-Container Garden (Kitchen Herbs)
Apr 1, 2015
DIANTHUS FLOWERS
Feb 27, 2015
Beauty of Flowers – Southern Gardening TV – July 24, 2013
Feb 26, 2015
Gardening Tips : Which Flowers Grow Well in the Shade?
Feb 24, 2015
Broccoli Flowers Attract Bees
Feb 23, 2015
#Red #Flowers #Roses #Italy #Pot #Rome #Plant #Vignette #Vintage #Green #Stone...
Feb 23, 2015
Laying Out Plants and Flowers [Flower Garden Landscaping Ideas]
Feb 22, 2015
FLOWERS in a NORTH EAST GARDEN
Feb 22, 2015
Summer Flowers in Ontario
Feb 22, 2015
Planting Flowers for Earth Day
Feb 22, 2015
Hardy Plants for May Flowers
Feb 21, 2015
Gardening Tips : How to Pinch Back Flowers
Feb 21, 2015
Different Kinds of Flowers and Trees in Florida
Feb 20, 2015
Gardening Tips & Flowers : Planting Herbs Indoors
Feb 20, 2015
PORTULACA PLANTS in BLOOM ~ PRETTY FLOWERS
Feb 20, 2015
Why Does My Trumpet Creeper Have No Flowers?
Feb 19, 2015
Female KiwiFruit Flowers – Difference Between Male Kiwi Fruit Flowers
Feb 19, 2015
Planting & Growing Annual Flowers : Planting Seasonal Annual Flowers
Feb 18, 2015
Vegetable Gardening & Plant Care : Caring for Fresh-Cut Flowers
Feb 18, 2015
Creating an Organic Container Using Herbs, Flowers and Roses
Feb 18, 2015
How-to Make Dried Flowers in a Shadow Box
Feb 17, 2015
Renee and P Allen Smith: Growing Flowers for Cutting
Feb 17, 2015
Male & Female Kiwi Flowers
Feb 17, 2015
DirectGardening’s One Cent Sale! Trees, Flowers, Seeds …
Feb 16, 2015
Flowers Resistant to Insects
Feb 16, 2015
When It Comes to Edible Flowers…
Feb 15, 2015
Stealing Marigold Flowers From My Wife, for My Vegetable Garden
Feb 15, 2015
How to Grow Plants & Flowers : How to Grow Edible...
Feb 15, 2015
TIPS for PLANTING WINDOW BOX FLOWERS
Feb 15, 2015
1
2
3
...
9
Page 1 of 9
STAY CONNECTED
1,808
Fans
Like
12,853
Followers
Follow
4,672
Followers
Follow
0
Subscribers
Subscribe
EDITOR PICKS
Hand Pollinating My Custard Apple (Aka. Sugar Apple, Cherimoya)
Jul 9, 2015
The New Landscape
gives you the best gardening tips, help, and advice including articles by top gardening experts, seasonal gardening advice and growing guide for hundreds of seeds, plants, flowers, and trees.
Contact us:
[email protected]
EVEN MORE NEWS
Tips on How to Trim a Cherimoya (Sugar Apple or Custard... | http://thenewlandscape.com/tag/flowers/ |
Q:
Missing Coordinates. Basic Trigonometry Help
please refer to my quick diagram attached below.
what i'm trying to do is get the coordinates of the yellow dots by using the angle from the red dots' known coordinates. assuming each yellow dot is about 20 pixels away from the x:50/y:250 red dot at a right angle (i think that's what it's called) how do i get their coordinates?
i believe this is very basic trigonometry and i should use Math.tan(), but they didn't teach us much math in art school.
alt text http://www.freeimagehosting.net/uploads/e8c848a357.jpg
A:
You don't actually need trigs for this one. Simply use slopes, or change in x and y.
Given a line of slope m = y/x, the line perpendicular to that line has slope -1/m, or -x/y.
The slope m between the red dots is -150/150, or -1/1. I noticed your positive y points down.
Therefore, the positive slope is 1/1. Both of your x and y changes at the same speed, with the same amount.
Once you know that, then it should be pretty easy to figure out the rest. Since they're aligned at 45 degrees angle, the edge ratio of the 45-45-90 triangle is 1 : 1 : sqrt(2). So if your length is 20, the individual x and y change would be 20/sqrt(2), or roughly 14 in integers.
So, your two yellow dots would be at (36, 236), and (64, 264). If the lines are not aligned to a convenient degree, you would have to use arctan() or something similar, and get the angle between the line and the horizontal line, so you can figure out the ratio of x and y change.
I hope my answer wasn't too hard to follow. For a more general solution, see Troubadour's answer.
Edit: Since the OP said the lower red dot is actually rotating around the upper red dot, we will need a more flexible solution instead.
I'm going to extend this answer from Troubadour's, since I'm doing exactly the same thing. Please refer to his post as you read mine.
1.
Get the vector from origin (200, 100) to rotating point (50, 250):
vector = (200 - 50, 100 - 250) = (150, -150)
2.
Rotate your vector by swapping the x and y, and negate x to get the new vector:
vector = (150, -150) => swap => (-150, 150) => negate x => (150, 150)
3.
Get the unit vector (of length 1) from the new vector:
vector = vector / length(vector)
= (150 / length(vector), 150 / length(vector))
~= (0.7071, 0.7071)
where
length(vector) = sqrt(150^2 + 150^2) ~= 212.2320
4.
Get the displacement vector of length 20, by multiplying the unit vector.
displacement_vector = vector * 20
= (0.7071 * 20, 0.7071 * 20)
= (14.1421, 14.1421)
5.
Add/Subtract this vector to/from your rotating vector (point):
yellow_1 = (50, 250) + (14.1421, 14.1421) ~= (64, 254)
yellow_2 = (50, 250) - (14.1421, 14.1421) ~= (36, 236)
I hope the above steps help you with formulating your code. Doesn't matter what the angle it's at, same steps.
A:
Call the red dot at ( 50, 250 ) A and the one at ( 200, 100 ) B.
One way would be to first calculate the vector AB i.e.
v_AB = ( 200 - 50, 100 - 250 ) = ( 150, -150 )
You can generate a vector at right angles to that by swapping the components and reversing the sign of one of the two components. So
v_AB_perp = ( 150, 150 )
is a vector rotated by rotating v_AB clockwise as you look at it on screen. You can normalise this to get a unit vector by dividing through by the magnitude i.e.
v_AB_perp_normalised = v_AB_perp / |v_AB_perp|
To get the yellow points just multiply this up by your 20 pixels and add/subtract it on to the co-ordinates of A.
| |
Ottoman fabric is an elastic fabric with a ribbed structure. Ottoman can be woven or knitted. When weaving the fabric, a thicker yarn is used for the weft yarn, which creates widthways ribbing. The distance and size of the ridges can differ per ottoman fabric. Ottoman has its origins in the Ottoman Empire, in present-day Turkey.
What is the difference between ottoman and corduroy?
Ottoman fabrics and corduroy both have a ribbed structure. Corduroy is a weave with a nap. The ribbing in corduroy is formed by raised, cut yarns. These yarns feel like tiny hairs and form a ribbed pile surface. Ottoman fabrics have no pile surface and therefore no nap. The ribs are woven into it.
Why ottoman?
The fabric is strong and tightly woven. Ottoman can be used for different types of clothing. The ribbed texture gives something extra to a plain fabric. Ottoman is used for everyday wear, formal wear and is known as coat fabric.
The ottoman fabrics at Knipidee contain some elastane. It makes the fabrics elastic and great for everyday clothing. | https://www.knipidee.nl/fabrics/ottoman |
How to paint a watercolor and ink flower...
4:07
Cells - Overview & Introduction
Other Resource Types ( 1,198 )
Lesson Planet
Reading Lessons: Townsend Press
Enhance your literacy unit with a set of videos featuring textual analysis and reading comprehension strategies. From discerning the main idea of a text to evaluating an author's purpose and tone, the ten-part series engages upper...
Lesson Planet
Reading Literature: 8th Grade ELA Common Core
Align your curriculum to the rigorous Common Core standards with a collection of reading lessons, activities, worksheets, and projects. From literary themes to annotation skills, the resources fold into any reading curriculum to ensure...
EngageNY
EngageNY Algebra II Module 4: Inferences and Conclusions from Data
The Algebra II Module 4 collection focuses on four topics: Probability, Modeling Data Distributions, Drawing Conclusions Using Data from a Sample, and Drawing Conclusions Using Data from an Experiment. Students show what they have...
Lesson Planet
Comprehension Strategies: Drawing Inferences
The proof is in the details! A richly detailed plan provides clear examples of how to draw inferences from text and how to provide support drawn directly from the text.
Lesson Planet
Guided Reading with Elizabeti's Doll
Practice reading strategies using Elizabeti's Doll by Stephanie Stuve-Bodeen. Readers utilize decoding and comprehension strategies before, during, and after reading the story. A detailed list of text features, high frequency words,...
Lesson Planet
Using Pre-reading Strategies: Infer
Use this resource to support your class practicing inference with poetry and visual art. The plan calls for an examination of "The Scream" by Edvard Munch and the "Mona Lisa" to promote speculation about artist's intent. From there, it...
Lesson Planet
Reading Comprehension: Guinness Book of World Records
If your learners are curious about human achievement, superlatives, or esoteric trivia, the Guinness Book of Records is a way to tap into instrinsic motivation and relevance. Here's an informational reading that will grab their attention...
Lesson Planet
The Big 7 Reading Strategies
Post these seven highly effective strategies used by good readers. The three-page resource details metacognition, schema, inferring, questioning, determining importance, visualizing, and synthesizing.
Lesson Planet
Reading Comprehension 2: Level 10
Are you a friggatriskaidekaphobic? An excerpt from an article about the fear of Friday the 13th is used as the basis of a reading comprehension exercise. The five questions require readers to employ several strategies (drawing...
Lesson Planet
Reading Comprehension 6: Level 9
Did you know that an acre of trees can absorb as much carbon dioxide as a car emits in 11,000 miles of driving? Such fun facts abound in a short reading comprehension passage detailing the benefits of parks and rooftop gardens. After...
Lesson Planet
Reading Comprehension 1: Level 9
The Iditarod Trail and the Iditarod Trail Sled Dog Race are the focus of a comprehension assessment. Readers must identify the main idea of the passage, draw inferences, define words using context clues, and identify the organizational...
Lesson Planet
Reading Comprehension 5: Level 10
Whether used as a reading comprehension assessment, as the basis of a mini-lesson on reading strategies, or as extra practice, this exercise will prove to be valuable because of the answers and explanation key that accompanies the...
Lesson Planet
Guided Reading: Asking Questions
Here is a reading strategies lesson in which learners use post it notes to create a bulletin board. They post their new questions on the bulletin board and look back at questions they have already learned the answer to. A great idea,...
Lesson Planet
Read with Me
A lesson boosts reading comprehension with strategies that focus on using text features to make predictions, inferences, share details, and decipher unknown vocabulary words. Pairs read a story together then discuss their most memorable...
Lesson Planet
Inferences Worksheet 2
You might infer that is worksheet is all about making inferences. And you'd be correct! Invite your learners to read four short passages. After reading each passage, pupils make inferences and support their inferences with textual evidence.
Lesson Planet
Reading Comprehension 1: Level 12
Need to assess the reading comprehension level of your class or of a new student? The subject matter of the reading passage, euthanasia, is sure to engage your readers’ interest while the questions test a variety of comprehension...
Lesson Planet
Inferences
A picture is worth a thousand words. Help your class out with some of those words with a video about drawing inferences from reading passages, images, and other media. The resource focuses on the supporting details of each passage as a...
Lesson Planet
Frindle: A Guiding Reading Unit
Guide your class through a reading of the popular children's book, Frindle, with this comprehensive literature unit. Starting with a brief introduction to the guided reading process, the class goes on to read the story two chapters at a...
Lesson Planet
During Reading Strategies (Inferring)
Pupils practice their reading skills. In this reading fluency and comprehension lesson plan, students read instructor-selected passages and use metacognitive skills to make inferences based on the selections.
Lesson Planet
Comprehension and Inference Question Creator
Encourage learners to ask questions about what they read with a worksheet about comprehension and inference questions. The resource provides directions and examples that guide kids through crafting their own questions.
Lesson Planet
What Do You See? (Inferences)
Making inferences is a skill that goes beyond the comprehension of written text. In this simple exercise, young learners are provided with a photograph and asked to answer a series of inference questions using only on the information...
Lesson Planet
Making Inferences (1)
Provide readers with an opportunity to practice drawing inferences by giving them this worksheet. Kids identify the text and author, record a sentence they believe infers rather than directly says, and then write the deeper meaning the...
Lesson Planet
Inferences (2)
Encourage young readers to use their prior knowledge, as well as text clues, to draw inferences from text. Provide them with this worksheet that asks them to record a passage, the background information they already have, the text clues...
Lesson Planet
Making Inferences (18)
Here's a bright idea. Model for readers how to use what they know about a story and combine this knowledge with clues from the text to formulate inferences about the story. | https://www.lessonplanet.com/search?keywords=reading+strategies+inferring |
The invention discloses a three-dimensional tailored thermal bonding invisible sock. The sock comprises a sock upper and a sock sole, the lower edge of the sock upper is connected with the edge of thesock sole in a sewn mode, a containing cavity is defined by the sock upper and the sock sole, a first hot melt adhesive film is thermally bonded to the upper edge of the sock upper in the circumferential direction and distributed on the inner side and/or the outer side of the sock upper, and a sock opening is defined by the upper edge of the sock upper and is oval. The sock sole part, corresponding to the foot sole, of the sock sole extends upwards to wrap the instep. The sock upper comprises a tiptoe coating part corresponding to the tiptoe, a sole coating part corresponding to the sole anda heel coating part corresponding to the heel, the tiptoe coating part, the sole coating part and the heel coating part are of an integrated structure, the cross section of the heel coating part is arc-shaped, the radian of the cross section of the heel coating part is the same as the radian of the section of the ankle of a human body, and an anti-skid piece is adhered to the hell coating part. The three-dimensional tailored thermal bonding invisible sock is comfortable to wear, durable, anti-falling and capable of preventing allergy. | |
Bumisuka.com – Mujair Fish Curry Recipe. Many people want to try the tilapia fish curry recipe. Processed fish does taste delicious and addictive. The savory and fresh gravy makes this dish a favorite.
How to make tilapia fish curry turns out to be very easy, the ingredients are not complicated, only kitchen spices are used every day. Come on, see and follow the steps of the following tilapia fish curry recipe.
- Dish Type: Main Menu
- Number of Servings: 2 servings
- Preparation Time: 15 minutes
- Cooking Time: 30 minutes
Mujair Fish Curry Recipe Ingredients
Prepare the following materials:
- 1 tilapia fish (400gr)
- 1 segment of galangal
- 1 lemongrass stalk
- 3 bay leaves
- 2 lime leaves
- To taste salt, sugar
- 1 tbsp coriander powder
- 1 tsp pepper powder
- 2 tablespoons oyster sauce (can be replaced with stock seasoning)
- 65 ml instant coconut milk
- Enough water
- 5 red onions
- 3 cloves of garlic
- 1 piece of ginger
- 1 segment of turmeric
- 2 hazelnuts
Mujair Fish Curry Recipe Cooking Tools
Prepare the following tilapia fish curry recipe cooking utensils:
- Receptacle
- Knife
- Wok
- Blender/cobek
- Plate
How to Make Mujair Fish Curry
Clean the fish then cut into pieces, sprinkle with salt and lime. Let stand for a while then fry half cooked. Set aside.
Prepare the spices, wash thoroughly. Galangal lemongrass in geprek. Then stir-fry the lemongrass, galangal, bay leaves, lime leaves until fragrant.
Add water, sugar, salt, oyster sauce / seasoning, cook until boiling. Then add the fish and coconut milk, cook until cooked. Taste correction. Lift.
Mujair Fish Curry is ready to be served.
How easy is it Sweet Couple? The delicious taste of fish and the right combination of spices will make anyone who eats it addicted. So, let’s immediately make this tilapia fish curry recipe at home! | https://bumisuka.com/mujair-fish-curry-recipe |
Where are the Most Maple Trees in Vermont?
September 20, 2022
Listen to the Podcast
If you’re wondering if there’s an epicenter of Vermont maple trees, look no further than rural Orange County. The county, home to the most maple trees in Vermont, is located east of the Green Mountains. The region includes Newbury, West Topsham, Vershire, Strafford, Thetford, Brookfield, Chelsea, Randolph, and Tunbridge.
One in every four trees in Vermont is a maple tree. Maple trees grow almost everywhere in Vermont, especially in Orange County.
-A classic Vermont fall scene / iStock photo
“I have to make reference to what you might call the ‘planetary epicenter’ of sugar maples, which is in Orange County, Vermont,” says Mike Snyder, Vermont Commissioner of Forests, Parks, and Recreation, in a Happy Vermont podcast interview. He explains that the geology of the Waits River drives much of the ecology in that particular area.
“Sugar maples happen there, and they happen in a glorious way,” Snyder adds. “They grow well, they do well, they’re healthy, and people have been taking good care of them there for a long time. It’s a really good place for sugar maples.”
Explore the Region with the Most Vermont Maple Trees on These Scenic Fall Drives
-Maple trees along a Vermont dirt road / iStock photo
If you’re looking for the most maple trees in Vermont, head to Orange County for scenic drives, small-town charm, and outdoor recreation.
Finding the Most Maple Trees in Vermont: Route 25 from Orange to Bradford
Route 25 extends for nearly 18 miles between the towns of Orange and Bradford along the scenic Waits River. This rural route features the iconic, widely photographed New Hope United Methodist Church in Waits River, as well as Lim Law Maple Farm in West Topsham and Northeast Slopes ski area in East Corinth.
If you want to get out into the woods, explore the Clement Loop Trail in Corinth. Afterward, head to the nearby Crossmolina Farm’s Cookeville Market, a four-season farm stand on Center Road.
Chelsea Mountain Road to Route 110
Chelsea Mountain Road, located between East Randolph in Chelsea, climbs along hillsides, meadows, and deep woods. At the end of the road, take a right on Route 110 south to Tunbridge, where you’ll find covered bridges, the North Tunbridge General Store, and the Tunbridge Fairgrounds.
The fairgrounds hosts the Tunbridge Fair in September and Vermont Sheep and Wool Festival in October.
-Cilley Covered Bridge in Tunbridge.
Stone Road to the Floating Bridge in Brookfield
Nothing beats traveling along a Vermont dirt road in the county with the most maple trees. On Stone Road, you’ll go past wide-open fields and coast under a broad canopy of trees. You’ll find the Floating Bridge crossing scenic Sunset Lake in Brookfield.
After you drive, walk or bike over Vermont’s only floating bridge, head west on Route 65 to Allis State Park in Randolph (day use is permitted off-season). Climb the park’s fire tower to soak up sweeping views of the Green Mountains.
-A view of the the Strafford Town House before the leaves turn.
Route 132 and Justin Memorial Highway in Strafford
The iconic Strafford Town Green alone will make you a lifetime fan of this Orange County community. The Town House, built in 1799, stands at the north end of the common between Brook Road and the Justin Memorial Highway. To the south is the Justin Morrill Homestead, a state historic site with public gardens, workshops, and events.
If you want to stretch your legs, explore Clover Wildlife Management Area, a 500-acre forested area open for hiking and wildlife viewing off Route 132. For a longer drive, take Route 132 east and continue to Thetford.
-An aerial view of maple trees in Vermont’s forests / Unsplash photo by Thomas Dils
Route 244 to Route 113 Between Fairlee and Thetford
From Route 5 in Fairlee, head west on Route 244 and wind around Lake Fairlee. Head south on Route 113 in Post Mills—home of the Vermontasaurus, created by the late Brian Boland—and to Thetford, a classic Vermont town.
Be sure to turn onto Academy Road and explore Thetford Hill State Park and Thetford Academy’s Woods Hill Trail and Torrey Mountain Bike Trail.
Route 5 in Newbury
Main Street in Newbury is listed on the National Register of Historic Places. It features everything you would want in a classic New England town—a general store, a town green, and easy access to outdoor recreation in the region with the most maple trees in Vermont. Along Route 5, you’ll find the Newbury Village Store, a 19th-century Methodist Church on the Town Green, and historic homes at every turn.
South of Newbury Village, head west on Snake Road to access Tucker Mountain Town Forest, one of Vermont’s newer town forests. The town forest offers public access for hiking, horseback riding, mountain biking, and snowshoeing, with panoramic views at the summit.
-Mike Snyder, Vermont Commissioner of Forests, Parks, and Recreation, in Corinth in 2020.
Happy Vermont Podcast: Fall Foliage, Where to Find the Most Maple Trees in Vermont, and the Future of Forests
In this episode of Happy Vermont, Vermont Commissioner of Forests, Parks, and Recreation Mike Snyder talks about this year’s fall foliage forecast. He also shares where to find the most maple trees in Vermont and what private ownership means for Vermont’s forests.
You can find Happy Vermont’s podcast on Spotify, Apple, iHeartRadio, Podbean, or wherever you listen to podcasts.
-Main Photo: An iconic scene in Waits River / iStock photo. | https://happyvermont.com/2022/09/20/where-are-the-most-maple-trees-in-vermont/ |
Friends reinvent Joan’s last Munro as a #hikeinthehoose
When friends of a keen walker Joan realised that lockdown would force her to cancel her last Munro this weekend, they decided to reinvent the event.
Elaine Shepherd has led the way in encouraging the Linlithgow Ramblers pals to climb the equivalent ascent of Beinn Na Lap at Corrour – but on their stairs at home.
Each walker is likely to climb their staircase hundreds of times to reach the total elevation of 562m (the height from the base of Beinn Na Lap to the summit at 937m.)
The walkers are also raising funds for during the lockdown event tomorrow (Easter Saturday).
Elaine said: “What started as a bit of a joke, ‘we should all climb our stairs on this day instead and raise funds for Scottish Mountain Rescue’ has quickly become a reality, so here we go. We’ll be all starting at 11.15am and we have no idea when we will finish.”
The ascent proper of Joan’s 282nd Munro in her full round has been postponed until October.
How many stairs?
If you would like to join the challenge or you want to understand what these friends will have to do, you need to measure the height of your stairs (for example, 19cm each), then multiply this number by the number of stairs (eg 14) and then divide the elevation height of Beinn na Lap (562m) by the height of your stairs
The Munro stair climb formula: 19X14= 266 (2.66metres), 562/2.66= 212 climbs (rounded up).
Elaine added: “Many hill walkers are missing being out in the hills at this time, although we appreciate how essential this is. Hill walking for many people isn’t just fitness and exercise, it’s their social life, too.
“As a very active walking group, we greatly appreciate the comfort and support that Scottish Mountain Rescue can bring and knowing that they are there if we need them is such a privilege.
“What we want to achieve most in this challenge is to make people smile, reconnect the walking community and have a bit of fun.
“We would like to encourage any other Ramblers, other walking groups or individuals to join in and would suggest a £10 donation to take part (not compulsory) and we’ll be sharing the challenge on Twitter and Facebook with the hashtag #hikeinthehoose.”
Make a donation to #hikeinthehoose
Already, the fun lockdown challenge has raised more than £900 for Scottish Mountain Rescue. You can also donate. | https://www.fionaoutdoors.co.uk/2020/04/friends-reinvent-joans-last-munro-as-a-hikeinthehoose.html |
Golden Gate Park uses 427 million gallons of SF drinking water per year. That's all about to change.
Sprinklers water a lawn at Golden Gate Park on June 14, 2021, in San Francisco.
Justin Sullivan/Getty Images
As I ease my body into child’s pose, my fingers stretch beyond the limits of my yoga mat, grazing the soft blades of grass of Hellman’s Hollow in Golden Gate Park. This sprawling meadow is one of my favorite expanses of green space in San Francisco's largest park, but as I moved through a series of sun salutations at a recent outdoor yoga class, I couldn’t take my eyes off that bright green, perfectly manicured grass.
Amid the year’s increasing drought, it must take a lot of water to keep that patch of land looking so pristine.
Turns out, it takes approximately 1.2 million gallons of water a day to keep the entirety of Golden Gate Park and the Panhandle looking so good. That number can fluctuate daily in response to the season and temperature, but it adds up to about 427 million gallons each year — which right now is all potable water.
The average household uses about 41 gallons of water per day in San Francisco, and the city delivers 36 million gallons of water to S.F. residents each day, meaning the upkeep of our biggest city park is no small feat.
Using drinking water to keep our parks luscious and green isn’t uncommon in San Francisco, but it’s something the city is working to change. Golden Gate Park currently uses its own groundwater wells to irrigate the park, which has water quality permitted for drinking water standards. The Panhandle uses water from the San Francisco Public Utilities Commission (SFPUC). By summer 2022, barring any delays, the water used for irrigation and lake fill would instead come from a recycled water system that’s currently under construction to replace the current watering system.
The project is designed to deliver up to 2 million gallons of water per day but could reach a peak demand of 4 million gallons of water per day.
This means in just a year from now, I’ll probably feel a lot better stretching out in Golden Gate Park to take a yoga class, knowing that the wastewater from my evening’s shower won’t head into the ocean (after wastewater treatment) but instead is irrigating the grass underneath my feet.
This massive project, dubbed the Westside Enhanced Water Recycling Project, has been underway since 2017 and will create a recycled water supply to use for nondrinking purposes. This conserves our current potable water from the city’s current system, and as California experiences more drought years, this could go a long way toward having reserves of drinking water that we may need down the road or in the event of a natural disaster like an earthquake.
But to do that involves a whole lot of construction. First, the agency built a new recycled water treatment facility at the Oceanside Treatment Plant. Then, construction began on almost 8 miles of new recycled water pipelines that snake underground up from the treatment plants through the streets of Parkside, the Sunset, Golden Gate Park and then the Richmond before stopping at the Presidio.
The future recycled water system in San Francisco.
SFPUC
There’s also the addition of an 840,000-gallon underground reservoir and an above-ground recycled water pump station in Golden Gate Park that will pump recycled water up to Lincoln Park, including the golf course, and the Presidio.
While “recycled water” seems an ambiguous term, it doesn’t mean the water’s not treated, as California still has some of the country’s highest standards for recycled water. Paula Kehoe, director of water resources at SFPUC, said the water will still go through technologies like microfiltration, reverse osmosis and ultraviolet light before it hits a blade of grass.
Reverse osmosis is a particularly important technology for the park since it removes salts, certain nutrients and ammonia from water. “There are a lot of salt-sensitive species in Golden Gate Park,” Kehoe said.
This high level of treatment will minimize the impacts the water could have on vegetation and species within the lakes. While there is some natural evaporation from the park’s lakes, there’s also some water that seeps out the bottom of the lakes, Kehoe explains, which is why they typically need to top them off every other year, though it depends on the lake.
When the first phase of the Westside Enhanced Water Recycling Project is complete in late summer or fall 2022, it will serve Golden Gate Park, Lincoln Park and the Panhandle. The next phase is expected to replace more than 70% of the San Francisco Zoo’s water use — right now about 70% of the zoo’s water usage is groundwater and 30% is from the SFPUC water system, Kehoe said. The zoo uses about 101 million gallons of water each year.
Then, it’s on to replacing much of the water usage at the Presidio, including the golf course and National Cemetery, which uses about 64 million gallons of water each year.
San Francisco isn’t the only city looking to up its recycled water usage. In LA, Griffith Park’s Wilson and Harding Golf Course (actually two 18-hole courses) already uses recycled water for irrigation. Irrigation for the LA Zoo's parking lot was switched to a recycled water system in 2009, a retrofit that saves enough potable water for 40 homes each year, according to a Los Angeles Department of Water and Power estimate.
It’s not just outdoor uses either. SFPUC wants to replace potable water with nonpotable water in nonpotable applications whenever they can. “We're constantly looking at different ways we can use water more efficiently,” Kehoe said. “San Francisco is the only city in the U.S. that has such an aggressive ordinance to require new developments and redevelopment projects to treat their water on site. We’re leading efforts across the country.”
Tessa is a Local Editor for SFGATE. Before joining the team in 2019, she specialized in food, drink and lifestyle content for numerous publications including Liquor.com, The Bold Italic, 7x7 and more. Contact her at [email protected]. | |
Are women held to a different standard than men in real-world evaluative situations? While lab-based evidence of double standards in evaluation exists, some have argued that competitive pressure in the market resolves any bias by creating a disincentive to favor candidates based on gender. However, even in this competitive setting where users are highly motivated to seek recommendations that yield the highest returns, this paper demonstrates that evaluators in a financial market setting do use gender to rank candidates (preferencing men) and that gender preferences led average-quality men to be given the benefit of the doubt while average-quality women were penalized. The researchers also disproved the wildly held belief that women are more risk-averse than men, observing the excess return volatility, expected return, investment horizon, and short position of investment recommendations to find that women and men are similarly risk-loving.
Evaluations take place frequently in the course of a person’s career (e.g. hiring, performance assessments) and they have a significant impact on economic outcomes. However, evaluations in real-world settings are more of an art than a science and when information is lacking or uncertain, evaluators often rely on other indicators to determine expected quality.
Research has shown that gender is associated with widely held beliefs about expected performance and is often used as one such indicator of quality.
Research has also shown that gender is used in evaluations even when information about the actual quality of an individual is known. This identifies the double standard bias that arises when women are held to a higher standard and indeed must outperform men (who, research has shown, are generally expected to be more competent) in order to receive similar evaluations and recognition. Thus, the standards by which men and women are judged differently has far-reaching effects and contributes to both social and economic inequality.
This paper uses a field-based study to determine whether double standards exist in real-world evaluations and explores when and how gender informs evaluations. The study used data from a private online knowledge-sharing platform used by buy-side investment professionals. Users of the platform submit investment recommendations and also evaluate the recommendations submitted by other users. In this competitive setting, users are highly motivated to select the best recommendations in order to earn the highest returns and the subsequent performance in the financial markets is unbiased.
The study found that users clicked on the submissions of recommenders with male names (e.g. Matthew) 33% more often than those with obviously female names (e.g. Mary). Average-performing men were preferred over similarly performing women and even in the top-performing quartile, men received more page views. In fact, women had to be the cream of the crop (in the top 10%) of users for their page views to equal those of men with a similar performance level. This finding provides evidence that women are indeed held to a double standard of evaluation.
In order to categorize by gender, each recommender was given a female name score from 0 (Matthew) to 99 (Mary), signifying how likely it was that the name belonged to a woman. Interestingly, the study found that even men with more feminine sounding names (and higher female name scores) received relatively fewer page views than their counterparts with more masculine sounding names. The study also observed users’ behavior after they have clicked on a submission and more information about the recommendation is provided.
Once evaluators have reviewed the submission in full, the feedback they provide reveals no significant difference between the evaluation of performance for male and female recommenders.
The researchers also examined whether some inherent difference in the female recommenders’ behavior may be influencing their evaluation. While ruling out any systematic gender differences, they paid special attention to risk-aversion behavior given its relevance to the financial services industry and the pervasiveness of perceived female risk-aversion. The study examined the excess return volatility, expected return, investment horizon, and short position of investment recommendations and declaratively showed that women and men act in similarly risk-loving ways.
Hiring and evaluation processes – Many organizations and companies are now exploring how to be gender-blind in their hiring and evaluation processes. For example, many symphony orchestras have adopted blind auditions where the musicians play behind a screen as they are evaluated – this has directly contributed to significant gains in female musicians’ representation in this highly competitive field.
Remove applicants’ names – Another practice many companies are increasingly adopting is to remove applicants’ names or use only initials when generating long-lists for consideration. While it is best practice to ensure that the short-list is intentionally diverse, the generation of the long-list in a gender-blind way can aid in this effort.
Recognize gender bias – This study demonstrates that even in the most competitive settings, gender bias still influences decision-making. Meritocracy cannot be assumed, even in situations where there is a financial disincentive to be gender-biased. Gender is considered a salient characteristic now in part because women are so underrepresented – increasing the representation of women at all levels could help to reduce this bias. | https://www.gendereconomy.org/double-standards-in-evaluation/ |
Masari seeks new revenue allocation formula for states, LGs
Governor Aminu Masari of Katsina State on Tuesday canvassed for increase in federal allocation to states and local governments, insisting that the current revenue allocation formula tilted more in favour of the Federal Government.
He maintained that states and local governments areas have more responsibilities, the development which he stressed, made them to require more percentage from the current revenue formula.
Masari made the declaration when a delegation from the Revenue Mobilisation and Fiscal Commission paid him a courtesy call as part of its members’ sensitization visit to the state.
The governor said, “From 1999 to date, we have not been able to come up with a new Revenue Allocation Formula despite constitutional provision to do so. The formula has become stagnant. It is lopsided. It puts more resources at the centre and less resources for states and local governments.
Read Also
“The burden and responsibilities of daily needs are at states and local government levels. The Federal Government has various revenue-earning parastatals and some of them do not contribute to the Federation Account but it is the consolidated revenue of the Federal Government and this revenue is not shared.
“The current formula has given the Federal Government over 50% while states and local governments share less than 40%. This is where the people are; this is where the problems are.
“This formula is stale, unfair, unjust and there has to be a change to address realities of today and prepare us for tomorrow.”
A Federal Commissioner at the RMAFC, Kabir Mashi, who led the delegation had earlier told the governor that his members were in Katsina on a sensitization exercise over a possible review of the existing vertical revenue allocation formula. | |
- 8 Why is the Euler-Mascheroni constant not a Liouville number?
- 8 Can I say "jalousie" instead of "store"?
- 7 Prove equivalence between $X$ Hausdorff and $X$ finite with discrete topology
- 6 Commutativity up to scalar implies commutativity in an algebra
- 6 "sans faille" ou "sans failles" | https://tex.meta.stackexchange.com/users/152296/user66288 |
Don't you find the idea of designing your own fabric fascinating? In this hand-printed tea-towel tutorial you will learn not only how to block print on fabric but also how to create your own geometrical stamps. No special skills needed to hand-print your own super-cute tea towels. It's super easy!
Supplies
You could also sew your own tea towels with a sewing machine. If you do, just use absorbent cotton or linen fabric. In any case, make sure your tea towels are washed, dried, and ironed before printing.
To get started, you need a big clean surface to work on, like a table top. Just throw down a plastic sheet to protect it from staining.
Step 1: Sketch Your Design
So first things first, define your design. Decide what shapes you want and how they will be laid out on your tea towel. I suggest you use simple geometric designs like triangles, circles and squares. It's good to have a general idea of your final product right from the start, but you can always fine tune it later on.
Step 2: Form Triangular Stamps From Erasers
Take the eraser and cut it diagonally. Make sure to keep your cutter perpendicular to the cutting surface.
Now you have two triangular pieces of eraser. Put them back together to form again the initial rectangular. This time cut the other diagonal.
At this point you should have four triangular pieces to make your stamping block.
Continue by cutting in the same way with the other two erasers. You'll end up with 12 individual triangular pieces in total.
Step 3: Create the Printing Block
By creating a stamping block you can print a repetitive pattern more effectively and efficiently. Take a piece of wood with a flat even surface. Now, take your triangular pieces and arrange them on your block. Play around with them until you are happy with the result. Remember to create a stamp that will help you get to the desired design of your towel.
Glue the pieces one by one into position. It will help if you start gluing them attached to the left-hand corner, so that you know where your design starts while printing.
Make sure to carefully aline your pieces for a perfect geometric result.
You can use any other flat and thick material as a base for your block, as long as it is stable and you can hold it easily. For example, I used a jam jar lid as a base for my stamp. To create an arrow, I glued together the two eraser arrows one mirroring the other and with two edges attached together.
Once you figure out how block printing works, you can try carving a more complicated design on a linoleum, rubber or cardboard block. If you try it, just remember that your design will be reversed when you print it!
Step 4: Make Circular Stamps From Potatoes
If you would like to create a circular stamp, an eraser is not the best carving material. Instead, you can achieve the circular imprint by using potatoes. The circle of course will not be geometrically perfect, but it will have a lovely organic shape. So take the potato and cut it twice with straight cuts perpendicular to the length of the potato.
If you cut only once you will end up with the same shape twice. With two cuts you have two different intersections of the potato, meaning two different sized circles. Keep the two end parts of the potato which are easier to hold. You don't really need the middle part of the potato for this project. (But you can always eat it!)
Step 5: Mix the Colours
If you want to create your own shades, mix the acrylic paint first. I used pale olive green and turquoise. Once you've sorted out the shades, mix the acrylic colour in the paint tray with the acrylic textile medium for printing in a 1:1 analogy (one part paint to one part medium). The medium gives fine colour definition and excellent colour fastness for more than twenty washes.
Step 6: Roll the Paint
Roll the paint evenly on a flat surface using the printing roller. You can do this inside the paint tray, if it has no lines.
Step 7: Apply Paint to the Stamp
Apply your paint on the stamping block by rolling with the roll or by using the foam paint brush. Try to apply the colour as evenly as possible on the stamp.
Step 8: Print the Fabric
First lay your towel on the printing surface. You can even stick it down with masking tape on all four sides to prevent it from slipping or wrinkling. Position the stamp in the correct spot and press it down.
For a repetitive design, check that your design is aligned with the existing printed design before placing the stamp down.
Make sure to press the stamp evenly on the fabric.
Keep on printing and have fun! When your towels are ready, clean your equipment with water.
Step 9: Let Dry and Iron
Let the towels dry for 24 hours. When dry, iron them to ensure the paint will stay put. Make sure you place piece of spare fabric between your printed design and the iron to prevent direct contact.
Step 10: It's Tea-Time!
I made six towels with six different geometric designs and I'm really happy with them! I'm sure your hand-printed tea towels will bring some art to your everyday life. Enjoy!
Have you made your own block prints before? Or is this your first time? Do you have any tips or questions? Let us know in the comments below.
Make Your Own Gorgeous Geo Hand-Printed Tea-Towels
Don't you find the idea of designing your own fabric fascinating? In this hand-printed tea-towel tutorial you will learn not only how to block print on... | https://www.diyhow.net/articles/make-your-own-gorgeous-geo-hand-printed-tea-towels.html |
What is a counteroffer?
A counteroffer is a new offer that is made in response to a previous offer that doesn’t meet the terms/requirements of the one the offer is being made to. Clear as mud, right? In a simpler term, it is a negotiation of terms between the buyer and seller.
In most cases this example is how it starts to play out, the buyer makes an offer to purchase a property from the seller. However, the seller wants to sell the property but doesn’t agree to all the terms laid out in the offer by the buyer, so the seller makes an offer back to the buyer – this is called a counteroffer.
What happens to the previous offer once a counteroffer is submitted?
When a counteroffer is made the previous offer is rejected and a new offer is created. Using the same example as above, the buyer is no longer obligated by the original offer terms to purchase the property from the seller once the new offer is created. Since the seller has technically rejected the first offer and made a new offer, the buyer now in turn has the option to accept, reject or counter the counteroffer.
Can you counter a counteroffer? If so, how many times?
If the buyer chooses to counter back, this in a sense becomes a counter-counteroffer; however, this is not a formal term that is used, so it is referred to just as another counteroffer. The back and forth process of offers and counteroffers can go on until either 1) an agreement to terms is accepted by both the buyer and the seller, 2) the last/current offer that is made is rejected by the opposite party with no new offer being made, or 3) the last party to make an offer withdraws the offer prior to its acceptance or rejection by the opposite party.
As negotiations go back and forth, it is always best to make sure to thoroughly review the new offer each and every time prior to making a decision on how to respond. There are many terms laid out in a contract, and even though it doesn’t always happen, reaching acceptance between all parties on all terms is the goal during this stage of a real estate transaction.
For additional resources on counteroffers, see the sites below for additional details. | http://www.easttexasrealestateonline.com/real-estate-vocabulary-101-counteroffer/ |
A polynomial is an algebraic expression of the form p(x) = + +……..+ + , where , , , are real numbers and ≠ 0 is called a polynomial.
The exponent of the highest power of the variable of a polynomial is called the degree of the polynomial.
If”x” is a variable in a polynomial (x), highest power of the variable x in a polynomial is called the degree of the polynomial.
The polynomial with all the coefficients as zero is called a zero polynomial.
A polynomial with a single term of a real number is called a constant polynomial.
A polynomial of degree one is called a first-degree or linear polynomial. the general form of a linear polynomial is ax + b, where a and b are real numbers and a≠0.
A polynomial of degree two is called a second-degree or quadratic polynomial. the general form of a quadratic polynomial is + bx + c, where a, b and c are real numbers and a≠0.
A polynomial of degree three is called a third-degree or cubic polynomial. the general form of a cubic polynomial is + + cx + d, where a, b, c and d are real numbers and a≠0.
A polynomial of degree four is called a fourth-degree or biquadratic polynomial. the general form of a biquadratic polynomial is + + + dx + e, where a, b, c, d and e are real numbers and a≠0.
The value of a polynomial p(x) when x = k (k is a real number) is the value obtained by substituting x as k.
It is denoted by p(k).
The zero of the polynomial is defined as any real value of x, for which the value of the polynomial becomes zero. | https://arinjayacademy.com/polynomials-class-10-maths/ |
When it comes to knowing where humans around the world actually live, resources come in varying degrees of accuracy and sophistication.
Heavily urbanized and mature economies generally produce a wealth of up-to-date information on population density and granular demographic data. In rural Africa or fast-growing regions in the developing world, tracking methods cannot always keep up, or in some cases may be non-existent.
This is where new maps, produced by researchers at Facebook, come in. Building upon CIESIN’s Gridded Population of the World project, Facebook is using machine learning models on high-resolution satellite imagery to paint a definitive picture of human settlement around the world. Let’s zoom in.
Connecting the Dots
Will all other details stripped away, human settlement can form some interesting patterns. One of the most compelling examples is Egypt, where 95% of the population lives along the Nile River. Below, we can clearly see where people live, and where they don’t.
View the full-resolution version of this map.
While it is possible to use a tool like Google Earth to view nearly any location on the globe, the problem is analyzing the imagery at scale. This is where machine learning comes into play.
Finding the People in the Petabytes
High-resolution imagery of the entire globe takes up about 1.5 petabytes of storage, making the task of classifying the data extremely daunting. It’s only very recently that technology was up to the task of correctly identifying buildings within all those images.
To get the results we see today, researchers used process of elimination to discard locations that couldn’t contain a building, then ranked them based on the likelihood they could contain a building.
Facebook identified structures at scale using a process called weakly supervised learning. After training the model using large batches of photos, then checking over the results, Facebook was able to reach a 99.6% labeling accuracy for positive examples.
Why it Matters
An accurate picture of where people live can be a matter of life and death.
For humanitarian agencies working in Africa, effectively distributing aid or vaccinating populations is still a challenge due to the lack of reliable maps and population density information. Researchers hope that these detailed maps will be used to save lives and improve living conditions in developing regions.
For example, Malawi is one of the world’s least urbanized countries, so finding its 19 million citizens is no easy task for people doing humanitarian work there. These maps clearly show where people live and allow organizations to create accurate population density estimates for specific areas.
Visit the project page for a full explanation and to access the full database of country maps.
Technology
Infographic: Generative AI Explained by AI
What exactly is generative AI and how does it work? This infographic, created using generative AI tools such as Midjourney and ChatGPT, explains it all.
Generative AI Explained by AI
After years of research, it appears that artificial intelligence (AI) is reaching a sort of tipping point, capturing the imaginations of everyone from students saving time on their essay writing to leaders at the world’s largest tech companies. Excitement is building around the possibilities that AI tools unlock, but what exactly these tools are capable of and how they work is still not widely understood.
We could write about this in detail, but given how advanced tools like ChatGPT have become, it only seems right to see what generative AI has to say about itself.
Everything in the infographic above – from illustrations and icons to the text descriptions—was created using generative AI tools such as Midjourney. Everything that follows in this article was generated using ChatGPT based on specific prompts.
Without further ado, generative AI as explained by generative AI.
Generative AI: An Introduction
Generative AI refers to a category of artificial intelligence (AI) algorithms that generate new outputs based on the data they have been trained on. Unlike traditional AI systems that are designed to recognize patterns and make predictions, generative AI creates new content in the form of images, text, audio, and more.
Generative AI uses a type of deep learning called generative adversarial networks (GANs) to create new content. A GAN consists of two neural networks: a generator that creates new data and a discriminator that evaluates the data. The generator and discriminator work together, with the generator improving its outputs based on the feedback it receives from the discriminator until it generates content that is indistinguishable from real data.
Generative AI has a wide range of applications, including:
- Images: Generative AI can create new images based on existing ones, such as creating a new portrait based on a person’s face or a new landscape based on existing scenery
- Text: Generative AI can be used to write news articles, poetry, and even scripts. It can also be used to translate text from one language to another
- Audio: Generative AI can generate new music tracks, sound effects, and even voice acting
Disrupting Industries
People have concerns that generative AI and automation will lead to job displacement and unemployment, as machines become capable of performing tasks that were previously done by humans. They worry that the increasing use of AI will lead to a shrinking job market, particularly in industries such as manufacturing, customer service, and data entry.
Generative AI has the potential to disrupt several industries, including:
- Advertising: Generative AI can create new advertisements based on existing ones, making it easier for companies to reach new audiences
- Art and Design: Generative AI can help artists and designers create new works by generating new ideas and concepts
- Entertainment: Generative AI can create new video games, movies, and TV shows, making it easier for content creators to reach new audiences
Overall, while there are valid concerns about the impact of AI on the job market, there are also many potential benefits that could positively impact workers and the economy.
In the short term, generative AI tools can have positive impacts on the job market as well. For example, AI can automate repetitive and time-consuming tasks, and help humans make faster and more informed decisions by processing and analyzing large amounts of data. AI tools can free up time for humans to focus on more creative and value-adding work.
How This Article Was Created
This article was created using a language model AI trained by OpenAI. The AI was trained on a large dataset of text and was able to generate a new article based on the prompt given. In simple terms, the AI was fed information about what to write about and then generated the article based on that information.
In conclusion, generative AI is a powerful tool that has the potential to revolutionize several industries. With its ability to create new content based on existing data, generative AI has the potential to change the way we create and consume content in the future.
Popular
-
Misc5 days ago
Visualizing the Odds of Dying from Various Accidents
-
Energy2 weeks ago
The Periodic Table of Commodity Returns (2013-2022)
-
Technology2 days ago
Infographic: 11 Tech Trends to Watch in 2023
-
Misc4 weeks ago
Infographic: The Longest Lasting Cars, in Miles
-
Politics2 weeks ago
Which Countries are the Most Polarized? | https://www.visualcapitalist.com/facebook-machine-learning-world-population-map/ |
Some Text Boxes throughout the program have spell checking built in. Words flagged as possible misspellings are underlined in red. When you see a red underlined word, right click on it.
Select the appropriate action: replace the word with a Hunspell generated suggestion, add it to the custom dictionary, or disable spell checking. If Disable Spell Check is selected, spell checking will remain disabled for the entire office until a user with the Security permission for Setup enables the Spell Check option.
Add to Dictionary
To manage the custom dictionary, in the Main Menu, click Setup, Spell Check.
You can enable/disable the spell check feature, add or remove words from the list, or modifiy an existing word. To modify a word, double click on it. | http://opendental.com/manual/spellcheck.html |
Last Saturday I came home early from office. As I entered my residence compound and reached my car parking bay I realized that today was not the same.
On any other day, I generally reach home late and always find cars parked both sides of my parking bay. And today after many months, I arrived early, to find the entire parking area completely vacant. This posed a unique problem.
Our parking lot is a bit narrow, we have to actually reverse our cars to park, but since everyday there were cars parked both sides I always had this reference point which was helped me park it in the right location and that too quickly. Though there are these yellow lines which designate each individuals parking bay in our society, but due to non maintenance they can hardly be seen. So when the other cars are parked, it served as a good reference point.
However today was a bit of struggle, since the bay lines were hardly visible and there was no other car parked on either side I had to do 3-4 iterations to get it right and to ensure its parked just right so that there is no inconvenience to both cars when they attempt to park later on. Once I did that, I collected my bag and just as I was heading towards the reception area, I looked back at my car once again to see if it had been parked right. And at that moment something struck which made me come home and write this blog (though posting it a week later!). Really don’t know whether it’s relevant, there may be loopholes in the logic but I really could connect with this
This parking lot experience taught me a something on entrepreneurship.
If you look at it there are two scenarios one encounters.
One when there are cars parked around the parking bay that provide a reference point to position your car. The key here is to simply use this reference point and ensure you park it just right. You don’t need to think too much. It’s much simpler. The success is dependent on how well you have parked in relation to the other two cars.
This scenario reminded me of the time when I worked in the corporate world. Just like how the cars that are parked on either side serve as reference points, in the corporate world too we encounter several such reference boundaries. These could be the corporate system one operates in or profile which is handed over at work. It could also represent expectation of one’s boss or simply colleagues around who you are constantly compared with. And as corporate individual we spend most of time trying to fit in within these reference points . Everything what we do is judged against this. We are given a profile which serve as one reference point then we have to work towards fitting in and do our best with it. We hardly think or challenge ourselves if we can go beyond it, as confirming to this profile is the most critical thing since we are judged based on it. Ditto with bosses, we constantly try and work towards ensuring that we conform to our bosses’ expectation out of fear for rejection. Our success lies in how well we can confirm to these reference point. One does not give much of a thought and all efforts are directed at ensuring one confirms to these reference point. You become more an expert of fitting it. How much does one grow in such a scenario?
The second scenario is very different. There are no cars parked in the bay, the lines are hazy. You have no reference point. You have to take your own call. You take 2-3 attempts to get it right. Yes some people get it right first time the others after more efforts. However once you do manage to successfully park it, the satisfaction is higher as you managed to do it and you then become the reference point for others.
This scenario is a lot more like entrepreneurship which is a lone journey. There are no reference points here unlike what you saw in the first scenario. No office systems, No bosses, No Job Profile, No colleagues / friends who you compare with. Because you have no reference point you have to chart your own course. You are forced to think different. With limited resources you give it a shot and take that decision. You may get it right or miss it first time. You try again and again. And after bit of struggle when you end up succeeding you not only feel satisfied that you have done something different but you actually become the reference point, a source of guidance to others around who want to do the same. And therein lies the biggest satisfaction of all.
So the next time you see a empty car parking lot (which is rare sight of course if you living in a metro in India) do give it a shot in exploring this connection of the parking lot with entrepreneurship
Perspectives
P.S All Images used in this post are for visual reference only. These are not actual depictions. | https://www.stevannoronha.com/what-my-parking-lot-taught-me-about-entrepreneurship/ |
Contact:
Background
One the most important impacts of human activity on herbaceous ecosystems is the application of fertilisers and the alteration of grazer-plant interactions.
Nash's Field experiment is situated in a section of grassland at the Silwood Park campus of Imperial College London. Since 1992, it has been manipulated to understand how the availability of nutrients and herbivory influence plant species composition and productivity, and how their effects impact other organisms and ecosystem functions.
This experiment has been supported by the Department of Life Sciences at Imperial College London and more recently by NERC grants to Dr Thomas Bell.
Experimental Design
The experiment is a factorial split-plot design, replicated in two blocks of 22 x 44m plots. The experimental plots receive a combination of treatments that include:
application of four nutrients in different combinations (N, K, P, Mg)
exclusion of vertebrates (rabbits)
exclusion of invertebrate herbivores (insects and molluscs)
manipulation of soil pH through periodic applications of lime
application of selective herbicides
Initiated by Professor Mick Crawley, treatments have been applied to all plots continuously since 1992. Data has been collected relating to the standing biomass or percentage coverage of each plant species present in all plots.
Key Results to Date
Rabbit herbivory is the primary factor determining plant community structure in Nash's Field grassland. Invertebrate herbivores have lesser impact but their effects are substantial enough to change the biomass of individual species and thus modify the structure of the community (Del-Val & Crawley, 2005). Exclusion of insects causes an increase in the dominance of grass species which in turn results in higher productivity but a loss of plant diversity. In contrast, the exclusion of molluscs, which feed selectively on seedlings of herbs, increase plant species richness but have no effect on biomass (Crawley, 2005, Allan & Crawley, 2011). Changes in plant community due to exclusion of invertebrate herbivores became apparent only after 8 years of the experiment (Allan & Crawley, 2011).
Exclusion of herbivores, particularly invertebrates, significantly decreases soil microbial biomass, community structure and abundance of microbial genes associated with biogeochemical cycles. These impacts appear to be driven by both the direct alteration of nutrient input into the soil and indirectly through changes in the plant community (Macdonald et al, 2015).
Fertilisation with a combination of nitrogen and phosphorus, both limiting nutrients in the soil of Nash's Field, results in an increase of plant biomass and a substantial reduction in plant species richness (Crawley, 2005). The addition of nitrogen also increases soil carbon sequestration when applied alone, but not when combined with potassium, magnesium and phosphorus. This effect is independent of soil pH or the composition of the plant community (Fornara et al, 2013).
Further Information and Data Availability
Further detail on the experimental design is available via the Silwood Park website.
Nash’s Field experiment is available as a research platform for ecologists to use. For access, consult the Silwood Park website or contact Dr Catalina Estrada.
To see a video of Thomas Bell explaining the importance of studying soil bacterial communities in the Nash’s Field experiment, click here. | https://www.ecologicalcontinuitytrust.org/nash-field |
It has been suggested that Range of a projectile be merged into this article. (Discuss) Proposed since May 2021.
Projectile motion is a form of motion experienced by a launched object. Ballistics (Greek: ?, romanized: ba'llein, lit. 'to throw') is the science of dynamics that deals with the flight, behavior and effects of projectiles, especially bullets, unguided bombs, rockets, or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance.
In a uniform gravitational field without air resistance, the horizontal and vertical components of velocity are independent from each other. Galileo Galilei dubbed this principle compound motion' in 1638 and used it to prove that the trajectory of a projectile is a parabola. The projectile's horizontal and vertical displacement as a function of time are:
where v0 is the initial speed, θ is the angle of the initial velocity with respect to the horizontal direction, and g is the downward gravitational acceleration.
Air resistance creates a force that depends on the projectile's speed through the medium. The speed-dependence of the friction force is linear () at very low speeds (Stokes drag) and quadratic () at larger speeds (Newton drag). The transition between these behaviours is determined by the Reynolds number, which depends on speed, object size and kinematic viscosity of the medium. For Reynolds numbers below about 1000, the dependence is linear, above it becomes quadratic. Qualitatively, the speed approaches a terminal velocity that depends on the drag and the particle's mass. The trajectory has a limited horizontal range, becomes vertically downward near this vertical asymptote, and reaches its maximum height lower and sooner than in the case of no air resistance.
Stokes drag, where , only applies at very low speed in air, and is thus not the typical case for projectiles. However, the linear dependence of on causes a very simple differential equation of motion
in which the two cartesian components become completely independent, and thus easier to solve. The solution in this approximation may be expressed in closed form as:
where , and the terminal velocity is . | https://popflock.com/learn?s=Projectile_motion |
These will be your fruiting branches.
Jul 13, When cutting any weeping tree, always cut out the"underneath" growth, leaving a branch or twig that is"springing" up in the right direction.
Here's a photo showing a branch that has been poorly pruned: the owner just wanted to make it shorter (it's a full-size weeping pear, not a miniature one) so they cut off the branch at random, leaving an ugly stump and a down-ward growing treedigging.club: Rachel The Gardener.
Nov 30, Follow the country cottage gardener as i prune and old weeping pear tree. Getting it back into shape ready for Spring. Re shaping the tree and cutting it bac. Jun 20, Pear tree pruning also begins at planting time. Cut back young, unbranched trees 33 to 36 inches ( cm.) above the ground to encourage good branching.
The cascading branches heighten the dazzling blossom show of pink, five-petaled single to double flowers which serve as an early food source for pollinators.
If your new tree has plenty of branches, remove those that are less than 18 inches ( cm.) from the ground and those with crotches of less than 60 degrees. How to Prune Pear TreesEstimated Reading Time: 2 mins. How to Prune a Weeping Pear Tree. Weeping pear trees grow in hardiness zones 4 to 7, and can reach a height and width of 20 feet. To keep the trees healthy and prevent branches from dragging on the ground, prune a weeping pear.
Sep 21, Pruning weeping trees is important because it improves the shape of the tree, keeps it healthy and promotes the circulation of sunlight and air. Prune weeping trees in the winter, before spring growing begins. Pruning in the fall doesn't give new growth time to become hardy enough to. Jun 12, Weeping Pear - how and when to prune? LizB62 Posts: 1. June in Plants. Evening (sorry - can't figure out how to search the forum, so there might already be a thread about this, but I can't find it.) I've got a tree in my front garden that I was told is a Silver Willow, and when I moved in, the tree was pruned into a very precise mushroom. | https://treedigging.club/weeping-pear-tree-pruning-strasburg-va.html |
In reply to:
Re: HOAG - name change to PRENTISS
1/26/00
My research shows the name was HOGG and was changed to Prentiss.Luther R. Prentiss married Abigail Patterson MCKINNEY on Jan. 6, 1837. She had 5 children from her first marriage (Jane,Nancy,Caroline,William & James).Was daughter of Nancy & Thomas Patterson.Have letters from Luther's parents dating early 1800's.Also many,many, many letters to Luther & his son Willard from his children with Abigail who were Zelma, Mendon, Mineda, Willard, MaryJane, & Ella. Am missing info on their last names as they only signed their first names to the letters.Luther had a sister Sarah (Sally) who married a Geo. Ayers.Have her letter to Luther up to the month he died. Luther's mother did not marry twice. Have letters from her up to near her death. Also have piece of Abner Hogg's knapsack & his promise to serve in the Revolutionary War.Happy to exchange info.
More Replies: | http://www.genealogy.com/forum/surnames/topics/prentiss/44/ |
What is Shanti Wellness?
As part of our ethos, we encourage our guests and staff to take a more conscious and proactive approach to their health, safety and wellness. With the help of our experts, we facilitate the refreshment and rejuvenation of the mind, body and soul to promote physical and emotional well-being.
How does Shanti Wellness work?
We aim to create a programme that reaches guests throughout the whole of their stay. From on-site Yoga in spas to well-being and mindfulness sessions in our kids' clubs, we can support guests of all ages and walks of life to ensure they leave us feeling refreshed and renewed.
Why is Shanti Wellness important?
We understand that our guests are looking for opportunities to develop a healthier lifestyle. The need for wellness-centred hospitality has never been more obvious than it is today, with more and more of our guests incorporating their own wellness routines into their daily lives. | https://www.shantihospitality.com/shanti-ethos/shanti-wellness/ |
This invention relates to a reference voltage generator used in a semiconductor integrated circuit.
Recently, elements in a semiconductor integrated circuit, especially in a dynamic RAM are being further miniaturized, with a result of lowering of a voltage suitable for deriving the transistor performance maximally. While, the power source voltage supplied to the dynamic RAM is not lowered. Therefore, an internal reduced-voltage generator is provided for lowering the voltage inside of the chip. In such a reduced-voltage generator, a reduced voltage based on a voltage generated at a reference voltage generator is supplied into the chip.
FIG.11 shows a conventional reference voltage generator (refer to Laid Open unexamined Japanese Patent Application No.63-244217). As shown in FIG.11, P-MOS transistors P100, P200 which are diode-connected from a power source V.sub.CC are provided in series at a first column and an N-MOS transistor N100 is connected in series between the P-MOS transistor P200 and a ground GND. At a second column, N-MOS transistors N300, N200 which are diode-connected from the ground GND are provided in series and a P-MOS transistor P300 is connected in series between the N-MOS transistor N200 and the power source V.sub.CC. The gate of the N-MOS transistor N100 and gate and drain of the N-MOS transistor N200 are connected to a node 200. The potential of the node 200 is the reference voltage V.sub.REF'. The gate of the P-MOS transistor P300 and gate and drain of the P-MOS transistor P200 are connected to a node 100. Accordingly, the output of the first column is inputted to the gate of the P-MOS transistor P300 via the node 100 to control the output of the second column and the output of the second column is inputted to the gate of the N-MOS transistor N100 via the node 200 to control the output of the first column, which is the feedback construction. For example, in FIG.11, the reference voltage output V.sub.REF' can be expressed by an equation (1), provided that all the MOS transistors are equal in gate length to one another and are operated in a saturation region. Wherein an absolute value of a threshold voltage of the P-MOS transistor is V.sub.TP, a mobility coefficient thereof is k'p, a threshold voltage of the N-MOS transistor is V.sub.TN, a mobility coefficient thereof is k'n, a gate width of the P-MOS transistor P300 is Wp, a gate width of the N-MOS transistor N100 is Wn, each gate width of the other MOS transistors is W, and a symbol * means multiplication. EQU V.sub.REF' =2*V.sub.TN *(1+2*(Wp*Wn/W.sup.2).sup.0.5)+2*(Wp*k'p/(W*k'n)).sup.0.5 *V.sub.TP( 1)
As indicated in the equation (1), the reference voltage output V.sub.REF' can set according to the gate width of each transistor, depends on the threshold voltages of the MOS transistors and is independent from the source voltage V.sub.CC. Also, the reference voltage V.sub.REF' can set according to the gate length (not indicated in the equation (1)). The condition for operating all the MOS transistors in the saturation region is that the power source voltage (V.sub.CC)>the set reference voltage output (V.sub.REF')-V.sub.TN +2*V.sub.TP, so that the reference voltage output V.sub.REF' is constant with reference to the power source voltage V.sub.CC.
In this way, the conventional reference voltage generator which easily determines the reference voltage according to the transistor size, using the threshold voltages of P-type and N-type MOS transistors, is independent from the power source voltage in a large range of the power source voltage but fairly depends on temperature. In detail, the threshold voltage of each of P-type and N-type MOS transistors which is to be the reference of the reference voltage generation depends on temperature and the absolute value thereof drops at high temperature. Therefore, as cleared from the equation (1), the reference voltage output V.sub.REF' drops at high temperature. Such the state is shown in FIG.12, which is a result of actual measurement of the conventional circuit manufactured. As cleared from the drawing, 3.30 V reference voltage output at 25.degree. C. deviates to 3.15 V at 100.degree. C. which means O.15 V (4.5%) drop to 75.degree. C. temperature change. Such the temperature dependency involves problems of lowering of the device speed at high temperature and increase of current consumption of the device at low temperature.
| |
There was a famous temple on a high hill in Assam. The priest of this temple was widely respected and was known to be a great scholar. When he grew old, he started searching for a younger priest who could take charge of the temple after his death. But, much to his dismay, he could not find any suitable person. As the priest lay on his deathbed, he called the trustee of the temple and told him, “After my death, make sure that only a ‘human being’ replaces me as the priest of this temple”. As soon as he said these words, he died. Information travelled far and wide that the head priest of the famous temple had died and now there was an urgent need for a replacement. A day was set for the selection of the successor. That day, starting at dawn, aspirants started trekking the steep and torturous climb to the temple. The route to the temple was indeed difficult; it was full of thorns, and stones. By the time most people managed to reach the temple, they had received minor cuts and bruises on their feet and hands.
Direction (Q. 1 - 3) : Choose the word / group of words which is most similar in the meaning to the word / group of words printed in bold as used in the passage.
Direction : Choose the word / group of words which is most similar in the meaning to the word / group of words printed in bold as used in the passage.
Direction (Q. 4 - 5) : Choose the word / group of words which is most similar in the meaning to the word / group of words printed in bold as used in the passage.
Direction (Q. 6 - 10) : In each question below, four words print in bold type are given. These are lettered 1., 2., 3. and 4.. One of these words printed in bold may either be wrongly spelt or inappropriate in the context of the sentence. Find out the word that is inappropriate or wrongly spelt, if any. The letter of that word is your answer. If all the words printed in bold are correctly spelt and appropriate in the context of the sentence then mark 4. i.e. ‘All Correct’ as your answer.
Q. 1) She fell ill due /2) to anxiety just /3) one week /4) before the ecsam.
Q. 1) They requested /2) everyone to take their /3) seats /4) and maintain silence.
Q. 1) They spoke /2) in such a laud /3) voice that even their neighbours /4) could hear them.
Q. 1) Manish accused /2) his rival /3) of steeling /4) his designs. | https://www.onlinetest.ibpsexamguru.in/questions/Clerk-English-Language/CE-Test-79 |
BACKGROUND OF THE INVENTION
SUMMARY OF THE INVENTION
DETAILED DESCRIPTION
1. Field of the Invention
The invention relates to a coordinate sensing system and a coordinate sensing method and, more particularly, to a coordinate sensing system and a coordinate sensing method utilizing a direction sensor and at least one magnetic sensor to sense a 3D coordinate of a magnetic member in a space. Specifically, the aforesaid coordinate sensing system and coordinate sensing method can be applied to a display system.
2. Description of the Prior Art
As motion control gets more and more popular, the present operation behavior of user may change in the future, wherein gesture control may be adapted for various applications. To recognize a gesture performed by a user in a space precisely, it needs to calculate a 3D coordinate of a device (e.g. presentation pen, game joystick, remote controller, etc.), which is operated by the user to perform the gesture in the space, precisely. The prior art usually utilizes a camera to capture an image of the user and then analyzes the captured image, so as to sense the 3D coordinate of the device in the space. The disadvantage of the aforesaid manner is that the camera must has high resolution and cooperate with complicated image processing algorithm. Consequently, it is necessary to use a high-level computation processor to execute the complicated image processing algorithm. Accordingly, the cost of an electronic product will increase.
Therefore, an objective of the invention is to provide a coordinate sensing system and a coordinate sensing method utilizing a direction sensor and at least one magnetic sensor to sense a 3D coordinate of a magnetic member in a space, so as to solve the aforesaid problems.
Another objective of the invention is to provide a display system equipped with the aforesaid coordinate sensing system and coordinate sensing method.
According to an embodiment of the invention, a coordinate sensing system comprises a magnetic member, a direction sensor, a first magnetic sensor and a processor, wherein the processor communicates with the direction sensor and is electrically connected to the first magnetic sensor. The magnetic member has a magnetic dipole moment. The direction sensor is used for sensing a direction of the magnetic dipole moment. The first magnetic sensor is used for sensing a first magnetic field of the magnetic member. The processor is used for calculating a first distance between the magnetic member and the first magnetic sensor and calculating a direction of the first distance according to a value of the magnetic dipole moment, the direction of the magnetic dipole moment and the first magnetic field. The processor is further used for calculating a coordinate of the magnetic member according to the first distance and the direction of the first distance.
According to another embodiment of the invention, a coordinate sensing method comprises steps of sensing a direction of a magnetic dipole moment of a magnetic member by a direction sensor; sensing a first magnetic field of the magnetic member by a first magnetic sensor; calculating a first distance between the magnetic member and the first magnetic sensor and calculating a direction of the first distance according to a value of the magnetic dipole moment, the direction of the magnetic dipole moment and the first magnetic field; and calculating a coordinate of the magnetic member according to the first distance and the direction of the first distance.
According to another embodiment of the invention, a display system comprises a display device and a coordinate sensing system. The display device is used for displaying an object. The coordinate sensing system comprises a handheld device and a coordinate sensing device. The handheld device comprises a magnetic member and a direction sensor. The magnetic member has a magnetic dipole moment. The direction sensor is used for sensing a direction of the magnetic dipole moment. The coordinate sensing device comprises a communication module, a first magnetic sensor and a processor, wherein the processor is electrically connected to the communication module and the first magnetic sensor. The communication module is used for communicating with the display device and the direction sensor. The first magnetic sensor is used for sensing a first magnetic field of the magnetic member. The processor is used for calculating a first distance between the magnetic member and the first magnetic sensor and calculating a direction of the first distance according to a value of the magnetic dipole moment, the direction of the magnetic dipole moment and the first magnetic field. The processor is further used for calculating a coordinate of the magnetic member according to the first distance and the direction of the first distance. The processor transmits the coordinate of the magnetic member to the display device through the communication module. The display device controls the object according to the coordinate of the magnetic member.
As mentioned in the above, the coordinate sensing system and the coordinate sensing method of the invention utilizes the direction sensor and at least one magnetic sensor to sense a 3D coordinate of the magnetic member in a space. In practical applications, the magnetic member and the direction sensor may be disposed in a handheld device including presentation pen, game joystick, remote controller, and so on, such that a user can operate the handheld device to perform a gesture to control an electronic device (e.g. display device) to execute specific function (e.g. eraser function, palm function, 3D drawing function, handwriting input function, cursor moving function, etc.).
These and other objectives of the present invention will no doubt become obvious to those of ordinary skill in the art after reading the following detailed description of the preferred embodiment that is illustrated in the various figures and drawings.
FIG. 1
FIG. 1
FIG. 1
10
12
10
12
10
10
12
10
12
10
12
x
y
z
Referring to , is a schematic diagram illustrating a magnetic member located at a random point P relative to a magnetic sensor in a space. As shown in , the magnetic member is located at the point P and the point O, where the magnetic sensor is located at, is an origin of the coordinates, wherein {right arrow over (m)} represents a magnetic dipole moment of the magnetic member at the point P and the direction of the magnetic dipole moment {right arrow over (m)} is parallel to a coordinate axis z of the Cartesian coordinate system; {right arrow over (B)} represents a magnetic field of the magnetic member sensed by the magnetic sensor relative to the point O; B, Band Brepresent three components of the magnetic field {right arrow over (B)} relative to the coordinate axes x, y and z; r represents the distance between the magnetic member and the magnetic sensor ; and θ represents an angle included between the distance r and the coordinate axis z (i.e. the angle included between the magnetic dipole moment {right arrow over (m)} and the radial direction {right arrow over (r)}). Therefore, the magnetic field {right arrow over (B)} of the magnetic member sensed by the magnetic sensor relative to the point O can be calculated by the following equation 1 in polar coordinates.
<math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mover><mi>B</mi><mo>⇀</mo></mover><mo>=</mo><mrow><mrow><mrow><msub><mi>B</mi><mi>x</mi></msub><mo></mo><mover><mi>i</mi><mo>^</mo></mover></mrow><mo>+</mo><mrow><msub><mi>B</mi><mi>j</mi></msub><mo></mo><mover><mi>j</mi><mo>^</mo></mover></mrow><mo>+</mo><mrow><msub><mi>B</mi><mn>2</mn></msub><mo></mo><mover><mi>k</mi><mo>^</mo></mover></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><msub><mi>μ</mi><mn>0</mn></msub><mrow><mn>4</mn><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>r</mi><mn>3</mn></msup></mrow></mfrac><mo></mo><mrow><mo>[</mo><mrow><mrow><mn>3</mn><mo></mo><mrow><mo>(</mo><mrow><mover><mi>m</mi><mo>⇀</mo></mover><mo>·</mo><mover><mi>r</mi><mo>^</mo></mover></mrow><mo>)</mo></mrow><mo></mo><mover><mi>r</mi><mo>^</mo></mover></mrow><mo>-</mo><mover><mi>m</mi><mo>⇀</mo></mover></mrow><mo>]</mo></mrow></mrow><mo>=</mo><mrow><mfrac><msub><mi>μ</mi><mn>0</mn></msub><mrow><mn>4</mn><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>r</mi><mn>3</mn></msup></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>3</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>m</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>θ</mi><mo></mo><mover><mi>r</mi><mo>^</mo></mover></mrow><mo>-</mo><mrow><mi>m</mi><mo></mo><mover><mi>k</mi><mo>^</mo></mover></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo>,</mo><mrow><mi>wherein</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mover><mi>i</mi><mo>^</mo></mover></mrow><mo>,</mo><mover><mi>j</mi><mo>^</mo></mover></mrow></mtd><mtd><mrow><mi>Equation</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mn>1</mn></mrow></mtd></mtr></mtable></math>
0
and {circumflex over (k)} represent three unit vectors of the coordinate axes x, y and z, μrepresents a permittivity of the air, and {circumflex over (r)} represents a unit vector of the distance r in a direction from the point P to the point O.
A component of the magnetic field {right arrow over (B)} relative to the coordinate axis z can be calculated by the following equation 2.
<math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mover><mi>B</mi><mo>⇀</mo></mover><mi>z</mi></msub><mo>=</mo><mrow><mrow><mfrac><msub><mi>μ</mi><mn>0</mn></msub><mrow><mn>4</mn><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>r</mi><mn>3</mn></msup></mrow></mfrac><mo></mo><mrow><mo>[</mo><mrow><mrow><mn>3</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>m</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>cos</mi><mn>2</mn></msup><mo></mo><mi>θ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mover><mi>k</mi><mo>^</mo></mover></mrow><mo>-</mo><mrow><mi>m</mi><mo></mo><mover><mi>k</mi><mo>^</mo></mover></mrow></mrow><mo>]</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mo></mo><mi>m</mi></mrow><mrow><mn>4</mn><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>r</mi><mn>3</mn></msup></mrow></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>3</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>cos</mi><mn>2</mn></msup><mo></mo><mi>θ</mi></mrow><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo></mo><mrow><mover><mi>k</mi><mo>^</mo></mover><mo>.</mo></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mi>Equation</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mn>2</mn></mrow></mtd></mtr></mtable></math>
The following equation 3 can be obtained by the equation 2.
<math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mn>3</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>cos</mi><mn>2</mn></msup><mo></mo><mi>θ</mi></mrow><mo>=</mo><mrow><mfrac><mrow><mn>4</mn><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>r</mi><mn>3</mn></msup><mo></mo><msub><mi>B</mi><mi>z</mi></msub></mrow><mrow><msub><mi>μ</mi><mn>0</mn></msub><mo></mo><mi>m</mi></mrow></mfrac><mo>+</mo><mn>1.</mn></mrow></mrow></mtd><mtd><mrow><mi>Equation</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mn>3</mn></mrow></mtd></mtr></mtable></math>
In magnetism, the magnetic field {right arrow over (B)} can be calculated in polar coordinates by the following equation 4.
<math overflow="scroll"><mtable><mtr><mtd><mrow><mover><mi>B</mi><mo>⇀</mo></mover><mo>=</mo><mrow><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mo></mo><mi>m</mi></mrow><mrow><mn>4</mn><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>r</mi><mn>3</mn></msup></mrow></mfrac><mo></mo><mrow><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>θ</mi><mo></mo><mover><mi>r</mi><mo>^</mo></mover></mrow><mo>+</mo><mrow><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>θ</mi><mo></mo><mover><mi>θ</mi><mo>^</mo></mover></mrow></mrow><mo>)</mo></mrow><mo>.</mo></mrow></mrow></mrow></mtd><mtd><mrow><mi>Equation</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mn>4</mn></mrow></mtd></mtr></mtable></math>
The following equation 5 can be obtained by the equations 3 and 4.
<math overflow="scroll"><mtable><mtr><mtd><mrow><mi>B</mi><mo>=</mo><mrow><mrow><mo></mo><mover><mi>B</mi><mo>⇀</mo></mover><mo></mo></mrow><mo>=</mo><mrow><mrow><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mo></mo><mi>m</mi></mrow><mrow><mn>4</mn><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>r</mi><mn>3</mn></msup></mrow></mfrac><mo></mo><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mn>3</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>cos</mi><mn>2</mn></msup><mo></mo><mi>θ</mi></mrow></mrow><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mrow><mo>=</mo><mrow><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mo></mo><mi>m</mi></mrow><mrow><mn>4</mn><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>r</mi><mn>3</mn></msup></mrow></mfrac><mo></mo><mrow><msup><mrow><mo>(</mo><mrow><mfrac><mrow><mn>4</mn><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>r</mi><mn>3</mn></msup><mo></mo><msub><mi>B</mi><mi>z</mi></msub></mrow><mrow><msub><mi>μ</mi><mn>0</mn></msub><mo></mo><mi>m</mi></mrow></mfrac><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>.</mo></mrow></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mi>Equation</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mn>5</mn></mrow></mtd></mtr></mtable></math>
It is assumed that
<math overflow="scroll"><mrow><mrow><mi>R</mi><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mo></mo><mi>m</mi></mrow><mrow><mn>4</mn><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>r</mi><mn>3</mn></msup></mrow></mfrac></mrow><mo>,</mo><mrow><mrow><mi>put</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mi>R</mi></mrow><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mo></mo><mi>m</mi></mrow><mrow><mn>4</mn><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>r</mi><mn>3</mn></msup></mrow></mfrac></mrow></mrow></math>
into the equation 5, and square the equation 5 such that the following equations 6 and 7 can be obtained.
<math overflow="scroll"><mtable><mtr><mtd><mrow><mi>R</mi><mo>=</mo><mrow><mfrac><mrow><mrow><mo>-</mo><msub><mi>B</mi><mi>z</mi></msub></mrow><mo>±</mo><msqrt><mrow><msubsup><mi>B</mi><mi>z</mi><mn>2</mn></msubsup><mo>+</mo><mrow><mn>8</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>B</mi><mn>2</mn></msup></mrow></mrow></msqrt></mrow><mn>4</mn></mfrac><mo>.</mo></mrow></mrow></mtd><mtd><mrow><mi>Equation</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mn>6</mn></mrow></mtd></mtr><mtr><mtd><mrow><msup><mi>r</mi><mn>3</mn></msup><mo>=</mo><mrow><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mo></mo><mi>m</mi></mrow><mrow><mi>π</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo>-</mo><msub><mi>B</mi><mi>z</mi></msub></mrow><mo>±</mo><msqrt><mrow><msubsup><mi>B</mi><mi>z</mi><mn>2</mn></msubsup><mo>+</mo><mrow><mn>8</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>B</mi><mn>2</mn></msup></mrow></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></mfrac><mo>.</mo></mrow></mrow></mtd><mtd><mrow><mi>Equation</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mn>7</mn></mrow></mtd></mtr></mtable></math>
The negative value is unnecessary such that the following equation 8 can be obtained by the equation 7.
<math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mi>r</mi><mo>=</mo><msup><mrow><mo>[</mo><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><mo></mo><mi>m</mi></mrow><mrow><mrow><mo>(</mo><mrow><msqrt><mrow><mrow><mn>8</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msup><mi>B</mi><mn>2</mn></msup></mrow><mo>+</mo><msubsup><mi>B</mi><mi>z</mi><mn>2</mn></msubsup></mrow></msqrt><mo>-</mo><msub><mi>B</mi><mi>z</mi></msub></mrow><mo>)</mo></mrow><mo></mo><mi>π</mi></mrow></mfrac><mo>]</mo></mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></msup></mrow><mo>,</mo><mi>wherein</mi></mrow><mo></mo><mstyle><mtext></mtext></mstyle><mo></mo><mrow><msup><mi>B</mi><mn>2</mn></msup><mo>=</mo><mrow><msubsup><mi>B</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>B</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><mrow><msubsup><mi>B</mi><mi>z</mi><mn>2</mn></msubsup><mo>.</mo></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mi>Equation</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mn>8</mn></mrow></mtd></mtr></mtable></math>
The aforementioned equations 1 and 8 will be used to describe the features of the invention in the following.
FIGS. 2 and 3
FIG. 2
FIG. 3
FIG. 2
FIGS. 2 and 3
3
3
3
30
32
30
300
32
320
322
320
3200
3202
322
3220
3222
3224
3226
3220
3222
3224
3226
3224
3220
3222
3220
30
3202
30
320
3200
3202
3222
3224
Referring to , is a schematic diagram illustrating a display system according to an embodiment of the invention, and is a functional block diagram illustrating the display system shown in . As shown , the display system comprises a display device and a coordinate sensing system . The display device is used for displaying an object (e.g. cursor). The coordinate sensing system comprises a handheld device and a coordinate sensing device . The handheld device comprises a magnetic member and a direction sensor . The coordinate sensing device comprises a communication module , a first magnetic sensor , a processor and a casing . The communication module , the first magnetic sensor and the processor are disposed in the casing . The processor is electrically connected to the communication module and the first magnetic sensor . The communication module may communicate with the display device and the direction sensor in wired or wireless manner, wherein the wired or wireless communication is well known by one skilled in the art, so it will not be depicted herein. In practical applications, the display device may be a liquid crystal display device, a smart TV, a projector or other electronic devices with display function; the handheld device may be a presentation pen, a game joystick, a remote controller or other devices capable of being operated by a user; the magnetic member may be a magnet or other members with a magnetic dipole moment; the direction sensor may be a gyro or other direction sensors; the first magnetic sensor may be a three-axis magnetic sensor (e.g. Hall sensor) or other magnetic sensors; and the processor may be a processor or controller with data processing function.
3200
3202
3200
3224
3220
3222
3200
3224
3224
3200
3222
3222
3224
3200
3224
3200
30
3220
30
300
3200
1
1
1
1
1
1
1
In this embodiment, the magnetic member has a magnetic dipole moment {right arrow over (m)}. The direction sensor is used for sensing a direction of the magnetic dipole moment {right arrow over (m)} of the magnetic member and transmitting the direction of the magnetic dipole moment {right arrow over (m)} to the processor through the communication module . The first magnetic sensor is used for sensing a first magnetic field {right arrow over (B)}of the magnetic member and transmitting the first magnetic field {right arrow over (B)}to the processor . Afterwards, through the aforementioned equations 1 and 8, the processor can calculate a first distance rbetween the magnetic member and the first magnetic sensor and calculate a direction of the first distance {circumflex over (r)}according to a value of the magnetic dipole moment m, the direction of the magnetic dipole moment {circumflex over (m)} and the first magnetic field {right arrow over (B)}. Then, by taking the first magnetic sensor to be the origin of the coordinates, the processor can calculate a coordinate of the magnetic member according to the first distance rand the direction of the first distance {circumflex over (r)}. Then, the processor can transmit the coordinate of the magnetic member to the display device through the communication module . Then, the display device can control the object to execute a predetermined function according to the coordinate of the magnetic member .
FIG. 4
FIG. 4
FIG. 4
3200
3202
3200
3202
Referring to , is a schematic diagram illustrating the magnetic dipole moment {right arrow over (m)} of the magnetic member parallel to the coordinate axis z of the Cartesian coordinate system. As shown in , when the direction of the magnetic dipole moment {right arrow over (m)} is parallel to the coordinate axis z of the Cartesian coordinate system, the direction sensor may be a two-axis gyro or other two-axis direction sensors. It should be noted that the aforesaid coordinate axis z may also be another coordinate axis x or y. Furthermore, when the magnetic dipole moment {right arrow over (m)} of the magnetic member is not parallel to any coordinate axis of the Cartesian coordinate system, the direction sensor has to be a three-axis gyro or other three-axis direction sensor.
FIG. 5
FIG. 5
FIG. 5
FIG. 5(A)
FIG. 5(B)
FIG. 5(C)
FIG. 5
32
30
3228
3226
3224
3200
3224
3200
30
3220
30
3200
320
1
3200
3228
3228
30
3200
1
3200
3228
30
30
Referring to , is a schematic diagram illustrating the coordinate sensing system used for controlling the display device to execute a predetermined function. As shown in , an indication plane may be defined on the casing . After the processor calculates the coordinate of the magnetic member , the processor can transmit the coordinate of the magnetic member and the direction of the magnetic dipole moment {circumflex over (m)} to the display device through the communication module . Then, the display device executes a predetermined function according to the coordinate of the magnetic member and the direction of the magnetic dipole moment {circumflex over (m)}. For example, when a user operates the handheld device to make an end E of the magnetic member face the indication plane (i.e. the magnetic dipole moment {right arrow over (m)} is toward the indication plane ), the display device will execute a predetermined function according to the coordinate of the magnetic member , such as an eraser function shown in , a palm function shown in , or a 3D drawing function shown in . It should be noted that when the end E of the magnetic member faces the indication plane , the predetermined function executed by the display device can be designed according to practical applications and can be set by the user. In other words, the predetermined function executed by the display device is not limited to the embodiments shown in .
FIG. 6
FIG. 6
FIG. 6(A)
FIG. 6(B)
FIG. 6(C)
FIG. 6
32
30
320
2
3200
3228
3228
1
2
3228
1
30
3200
320
2
2
3200
3228
2
30
3200
320
3
2
3200
3228
3
3222
3200
30
2
3200
3228
30
30
Referring to , is another schematic diagram illustrating the coordinate sensing system used for controlling the display device to execute a predetermined function. As shown in , when a user operates the handheld device to make an end E of the magnetic member face the indication plane (i.e. the magnetic dipole moment {right arrow over (m)} is away from the indication plane ) and a distance d between the end E and the indication plane is smaller than a first threshold (e.g. d is smaller than 3 cm), the display device will execute a predetermined function according to the coordinate of the magnetic member , such as a handwriting input function. As shown in , when the user operates the handheld device to make a distance d between the end E of the magnetic member and the indication plane be larger than a second threshold (e.g. d is larger than 5 cm), the display device will execute a predetermined function according to the coordinate of the magnetic member , such as a cursor moving function. As shown in , when the user operates the handheld device to make a distance d between the end E of the magnetic member and the indication plane be larger than a third threshold (e.g. d is larger than 10 cm), the first magnetic sensor cannot sense the magnetic member such that the display device will enter an idle state. It should be noted that when the end E of the magnetic member faces the indication plane , the predetermined function executed by the display device can be designed according to practical applications and can be set by the user. In other words, the predetermined function executed by the display device is not limited to the embodiments shown in .
FIG. 7
FIG. 7
FIG. 7
32
30
100
3200
3202
102
3200
3222
104
3200
3222
106
3200
108
3200
30
110
30
3200
1
1
1
1
1
1
Referring to , is a flowchart illustrating a coordinate sensing method according to an embodiment of the invention. The coordinate sensing method shown in can be implemented by the aforesaid coordinate sensing system and display device . First of all, step S is performed to sense a direction of a magnetic dipole moment {circumflex over (m)} of a magnetic member by a direction sensor . At the same time, step S is performed to sense a first magnetic field {right arrow over (B)}of the magnetic member by a first magnetic sensor . Afterwards, step S is performed to calculate a first distance rbetween the magnetic member and the first magnetic sensor and calculate a direction of the first distance {circumflex over (r)}according to a value of the magnetic dipole moment m, the direction of the magnetic dipole moment {circumflex over (m)} and the first magnetic field {right arrow over (B)}. Step S is then performed to calculate a coordinate of the magnetic member according to the first distance rand the direction of the first distance {circumflex over (r)}. Step S is then performed to transmit the coordinate of the magnetic member and the direction of the magnetic dipole moment {circumflex over (m)} to the display device . Finally, step S is performed to execute a predetermined function in the display device according to the coordinate of the magnetic member and the direction of the magnetic dipole moment {circumflex over (m)}. It should be noted that the predetermined function is described in the aforesaid embodiments, so it will not be depicted herein again.
FIGS. 8 and 9
FIG. 8
FIG. 9
FIG. 8
FIGS. 8-9
FIGS. 2-3
3
3
3
3
322
32
3230
3226
3224
3230
Referring to , is a schematic diagram illustrating a display system ′ according to another embodiment of the invention, and is a functional block diagram illustrating the display system ′ shown in . The main difference between the display system ′ and the aforesaid display system is that the coordinate sensing device ′ of the coordinate sensing system ′ further comprises a second magnetic sensor , which is disposed in the casing and electrically connected to the processor . In practical applications, the second magnetic sensor may be a three-axis magnetic sensor (e.g. Hall sensor) or other magnetic sensors. It should be noted that the same elements in and are represented by the same numerals, so the repeated explanation will not be depicted herein again.
3230
3200
3224
3224
3200
3230
3230
3224
3200
3200
2
2
2
2
2
2
2
The second magnetic sensor is used for sensing a second magnetic field {right arrow over (B)}of the magnetic member and transmitting the second magnetic field {right arrow over (B)}to the processor . Afterwards, through the aforementioned equations 1 and 8, the processor can calculate a second distance rbetween the magnetic member and the second magnetic sensor and calculate a direction of the second distance {circumflex over (r)}according to the value of the magnetic dipole moment m, the direction of the magnetic dipole moment {circumflex over (m)} and the second magnetic field {right arrow over (B)}. Then, by taking the second magnetic sensor to be the origin of the coordinates, the processor can calculate the coordinate of the magnetic member according to the second distance rand the direction of the first distance {circumflex over (r)}. In other words, the invention may utilize another magnetic sensor to assist in calculating the coordinate of the magnetic member , so as to enhance the accuracy of the sensed coordinate.
Compared with the prior art, the coordinate sensing system and the coordinate sensing method of the invention utilizes the direction sensor and at least one magnetic sensor to sense a 3D coordinate of the magnetic member in a space. In practical applications, the magnetic member and the direction sensor may be disposed in a handheld device including presentation pen, game joystick, remote controller, and so on, such that a user can operate the handheld device to perform a gesture to control an electronic device (e.g. display device) to execute specific function (e.g. eraser function, palm function, 3D drawing function, handwriting input function, cursor moving function, etc.).
Those skilled in the art will readily observe that numerous modifications and alterations of the device and method may be made while retaining the teachings of the invention. Accordingly, the above disclosure should be construed as limited only by the metes and bounds of the appended claims.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1
is a schematic diagram illustrating a magnetic member located at a random point relative to a magnetic sensor in a space.
FIG. 2
is a schematic diagram illustrating a display system according to an embodiment of the invention.
FIG. 3
FIG. 2
is a functional block diagram illustrating the display system shown in .
FIG. 4
3200
is a schematic diagram illustrating the magnetic dipole moment of the magnetic member parallel to the coordinate axis of the Cartesian coordinate system.
FIG. 5
is a schematic diagram illustrating the coordinate sensing system used for controlling the display device to execute a predetermined function.
FIG. 6
is another schematic diagram illustrating the coordinate sensing system used for controlling the display device to execute a predetermined function.
FIG. 7
is a flowchart illustrating a coordinate sensing method according to an embodiment of the invention.
FIG. 8
is a schematic diagram illustrating a display system according to another embodiment of the invention.
FIG. 9
FIG. 8
is a functional block diagram illustrating the display system shown in . | |
You read that right: Ford Motor Company, for the first time since perhaps 1937, employs more hourly American workers in its factories than General Motors. Ford has been hiring an average of 10 union workers per day since 2011, according to Automotive News.
Granted, a substantially larger quantity of Ford’s production occurs within US factories, including (notably) the F-150 full-size pickup, which itself recently compelled the automaker to create 1,550 new factory jobs. In percentage terms, 78 percent of Ford’s North American production took place in US factories last year, vs. 61 percent for General Motors. The ratio likely becomes even more skewed when other continents are taken into account.
Still, Ford Motor Company employs about 50,703 factory workers here in the US (as of February 1st), compared to GM’s 50,300 or so. Said General Motors Spokesman Bill Grotz, “It’s all market-driven. We build to market demand. The head count hasn’t fluctuated that much over the past few years, but we’ve invested a lot of money in our operations.”
Or about $11 billion since 2009, if it interests you. | https://gmauthority.com/blog/2015/02/ford-motor-company-beats-gm-in-number-of-american-factory-workers/ |
Search the Community
Showing results for tags 'stocking'.
Found 78 results
-
Stocking experiment
Rod posted a topic in Ponds And AquaponicsI've just established a New pond (triangle shape 4mx2mx400mm deep) in my front courtyard and have decided to experiment stocking it Current stock(all fish either wild caught or juvies I have breed) White Clouds Tiger Endlers Blue eyes(from Conondale) Rainbows(from Conondale) Firetail gudgeon(from Caboolture) Pair Kribensis Pair Blue-eye cichlids Pair Paradise fish 50 odd glass shrimp 6 mystery snails dose of black worms Fish have been introduced gradually over last 6 weeks All going well....cichlids in breeding colours Will be interesting to see how each species fare....what will be there next year??
-
800l-1200l indoor pond stocking
catfishcrazy posted a topic in General Aquarium DiscussionHi, I'm new to the forum I have had an Australian bass for about 3 years and I need to upgrade I've found a range of ponds from 800l-1200l I plan to stock: 2 x Archerfish 1 x Snakehead gudgeon 1 x Australian Bass I'm massive fan of natives and I am wondering if there is anything that I could add? It doesn't matter if it isn't native but that would be preferred The pond will be heated, and the 1200 litre's dimension are 175cm x 118cm x 60cm Thanks, catfishcrazy
-
Help with 138g stocking
TheFuzzyAussie posted a topic in General Aquarium DiscussionHey guys, I'm a serial mind-changer and I can't make up my mind for my 6ft 138g tank stocking. Dimensions are 183cm long x 48.5cm deep x 60.5cm tall. Filtration is either a FX6 or sump, still undecided. I've gone through several phases: a full community tank, a rainbow tank, a discus tank, a tanganyikan tank, and mbuna tank or a 'monster' tank. Yet I still can't make up my mind... SO I wanna hear your guys suggestions. If you had this tank what would you want to stock it with? I'd love to hear any and all suggestions! I look forward to hearing them all!!! - Fuzzy Sent from my SM-G920I using Tapatalk
-
best size cube and stocking ideas.
RJHELL posted a topic in General Aquarium Discussioni cant decide what size cube to go. 2ft, 2.5ft or 3ft. love a 3ft but cleaning could be abit of a problem. 2ft is looking abit small. im thinking 2.5ft would be the best size. what is peoples ideas? the next problem would be stocking ideas, throw them in too.
-
Stocking advice
Kalo1993 posted a topic in General Aquarium DiscussionOkay so recently i have set up a pool/pond under my house that holds around 900-1000 gallons. It will be unheated and i plan on stocking it with: 1 longfin eel 1 coal grunter 1 golden perch 1 aussie bass 1 sleepy cod 1 tandanus catfish 1 pleco (maybe) 1 black shark (maybe) and a school of silver dollars over the warmer months. How do you think this mix will go? any other suggestions?
-
Hi, I recently bought an aquarium and would like some advice on how to stock it. I like the look of cichlids but want it to be a planted aquarium, any tips? I am leaning towards picking discus.
-
New 6x2x2 stocking ideas
rennie_08 posted a topic in General Aquarium DiscussionHi guys, I have recently set up a build that I started a long long time ago due to not having the time to finish it sooner with work! It now has water in it and now the partner and I can't decide what we want to put in there. The tank has black quartz gravel and a black background and we want to put driftwood and either real or fake plants to make the colour. My partner was thinking discus but I don't like the idea of keeping a tank at 30 degrees throughout winter and getting a power bill.... I suggested a peacock display? can anyone help suggest a stocking list with plenty of color? Filtration is 2 fluval fx5s. thanks in advance chris
-
Neon tetra and tiger barbs
Brettyd posted a topic in General Aquarium DiscussionHi everyone very new to aquariums and been doing a lot of research on what to put in my new tank. 200L tank. I like tiger barbs and neon tetra but just wondering will the barbs just be too aggressive?
-
Lake Tanganyika Stocking options 240L?
♡Sukha♡ posted a topic in General Aquarium DiscussionThoughts on this community Lake Tanganyika list? Cyprichromis Leptosoma x 16 Julidichromis Regani x 3 Xenotilapia Ochrygenys x 6 Tanganicodus Irsacae x 6 Neolamprologus Multifasciatus x 6 Synodontis Petricola x 5
-
6x2x2 stocking advice
cblaxall posted a topic in General Aquarium DiscussionHi all, avid reader of the forum, but never really participate (just in the background soaking up advice!) I have a new 6x2x2 aqua one aquience aquarium all set up and nearly finished fishless cycling, set up with a FX5 so plenty of filtration. I'm looking for stocking suggestions along the lines of s.a. cichlids . Going into the tank will be a 10cm BGK and a 15cm sailfin pleco (both from one of my other tanks) With this in mind, what do you suggest? Open to all ideas, I'll do plenty of reading regardless. Also open to sales if anyone has something of interest, I'm in the Logan area. Thanks and compliments on such a helpful, friendly community. Chris.
-
Stocking Suggestions - 3 ft Community Tank
FishTastic posted a topic in General Aquarium Discussion.Hi everyone. I think I might finally take the plunge and get my first 'big' tank. Something around the 3ft mark or a little bit bigger. I need some help with the stocking. So far I am thinking: 1 x Coral Blue Dwarf Gourami Neon Tetras (how many?) Corydoras (how many/what type?) Kuhli Loaches (how many?) ^^^^ Those are the fish I am really set on having. I would also like to have some guppies and/or mollies, but I am not overly worried whether I can have them or not. I think I have heard that mollies are extremely aggressive. Is this true? I will do sand for the substrate and somewhat planted (combo of fake and real, depending on what I can keep alive). Any suggestions for what else I can have? Will they get along? Will I be overstocked if I get all of these fish?
-
New 55g Cichlid tank stocking Questions
swys42 posted a topic in General Aquarium DiscussionHi Guys, Ive been researching alot for good colour in a cichlid tank and compatibility for 55g/4feet/230l tank and this is what i came up with. if you could tell me what you think and if they will fight for any reason. 5 x Red Zebra 5 x Pseudotropheus Salousi 5 x Rusty Cichlid 5 x Cynotilapia Afra Cobwe [h=1][/h]My only worry is the Afra and Soulousi, they might be too similar and fight?. Ill be swapping males for females once the get older and can easily be sexed. The tank will have a 2200LP/H filter so no problems in overstocking Thanks Francois
-
Growth rates and stocking
frannles posted a topic in General Aquarium DiscussionSo I'm slowly converting to larger natives and been buying some fingerlings as I enjoy watching them grow. Currently I have the following: 1 Murray cod 5 cm 1 barra 12 cm 1 jade perch 30 cm 7 bass 7 cm 4 tandanus cats 12 cm 1 saratoga lei 10 cm At the moment the barra and the perch are in a 4x2, cats and bass in a 3 ft and the saratoga and the cod in separate 3 ft tanks. They'll be moved to larger tanks once they get to 20cm or so. The long term plan is to have the cod in its own outdoor pond and to have 2 8x3x2 ft tanks for display. My main concern is that they'll all have different growth rates that they can't be moved at the same time to bigger tanks in a year or so and end up having more tanks running..... is there anyway to try and have them grow at a similar rate? From my understanding bass are slow growers and everything else is fast Also should I be putting them together when they're small so that they'd be less agro to each other when they're big? And with the current stock list and future plan would I be able to get more fish now or should I just leave it? I haven't fully decided on which fish are going with each other yet so ideas and opinions would be great
-
Im new so sorry if this is in wrong section.. Im in the process of planning my first low light planted tank (I currently have 150L fancy goldie tank) Tank specs: Aqua one aquience 1200rt 320L 120cmx50cmx65cm 1250 Aquis external canister filter 1400L/hr = 4.3x tank volume 2x 150watt heaters 4x 39 watt T5 bulbs (1.8watts per gallon) Ideal water parameters: (from my research on below stocking plan and aqadvisor) Ph: 6.5-7.5 Water hardness : 5 - 15dH Thermostate set to 22C Proposed stocking idea: 6x sterbai cory 1x bristlenose 8x rosy barbs 5x male swordtails 8x galaxy rasboras 8x danios 5x dwarf gourami (2x male 3x female) Im not sure about the dwarf gourami numbers though in this size tank so any advice would be great? Aqadvisor says im 98% stocked
-
fish stocking numbers
craig e posted a topic in General Aquarium Discussionhi all was wondering if my tank is overstocked? my tank measures 48l x14w x20h and the fish I have in it are as follows, 4 black widow tetras,1 silver dollar,3 angels,2 serpae tetras,11 neons,8 rummy nose,2 red eye tetras,5 mollies,3 platys,5 peppered catfish & 2 bristle nose catfish.also was wanting to add maybe 10/15 guppies so was wondering which fish I should cull to add these?using these tank measurements how many guppies alone could I stock in this tank? any feedback will be muchly appreciated cheers craig
-
Stocking a 8'x2'x2' tank with 6'x18"x18" sump
Kitah posted a topic in General Aquarium DiscussionI was just wondering if I can get some opinions on stocking an 8x2x2ft tank which will be filtered by a 6'x18"x18" sump, most likely with 6000lph return pump (two eheim compact 3000's) I currently have a 6x2x2 running two canisters, with weekly water changes that stocks 4 oscars, 2 sailfin plecos, 1 banded leporinus and 2 rivulatus. My fish are pretty good, very, very rarely does anyone injure anyone else- my two red tiger O's have paired up, as have my two lutino red o's- they defend their buddy, and otherwise leave the other fish along and just swim around together, in their respective pairs. Usually the only bickering I've ever had was when I had 4 rivs, the rivulatus seem to have little quarrels more frequently but rarely to the extent of injuring each other badly. The rivulatus leave the oscars alone, and visa versa. Fish I was wondering about (though I wouldn't get all of these) - dithers e.g. silver sharks, spotted silver dollars, tinfoil barbs? would these contribute to the tank by reducing aggression at all? - Blue acara - Severums- probably green, or possibly 2 green 2 gold, or just 1 of each - bocourti - 2 more rivulatus (one gold one white) - 2 albino sailfin plecos - uaru - braziliensis If these were compatible, what sort of numbers would be sustainable with this setup? I usually do water changes every 1-2wks, never over 2wkly. Edit: actually while I'm posting/asking anyway.. what are your thoughts on heaters? I always liked the idea of the jagers, but two of the last ones I bought ended up with moisture inside the heaters themselves- and no they were never dropped or anything. Has anyone used the shogun heaters? and would I be best with two 300w- would they be enough? vs. one massive heater... All suggestions are appreciated
-
Undecided stocking
lgw posted a topic in General Aquarium DiscussionI have a 5'x18"x20" tank with 4'x15"18" sump which I am in the process of restocking. I have a some ideas so far and would like some opinions/ideas on which I should go for. Feel free to share pics or previous experience with similar set ups. 1. All male peacock ( with possibly a few haps ) 2. Peaceful South American; tapajos, festivum, tetras, severums etc? 3. 'Peaceful' Central American; mid sized cichlids with some sword tail dithers? 4. Mbuna
-
SPONSERS stocking seachem products ?
th3f0rg0t3n posted a topic in General Aquarium DiscussionWhich of the forum sponsers sell SEACHEM products ?? A quick look on aquaholics,age of aquariums,the tech den, shows none of those 3 carry the products... Alternatively where do you buy your SEACHEM products, thereefshop,guppys or local LFS ??
-
Ideas for stocking a planted tank.
BarryToucher posted a topic in General Aquarium DiscussionHey guys, I've just had a mass extinction of my neon tetras in my planted tank(too much co2) I've already got a Central American cichlid display tank as well as an African cichlid display tank.. what too next? i don't want to go tetras again, i want something with personality. ph is low at 5.5-6 but hardness is still an issue as the tap water here is very hard and alkaline. was thinking dwarf cichlids, but have heard they are quite difficult to look after? was also thinking Rainbows and other natives. throw me some ideas guys, feeling a little down after my mistake, lost 60 neon tetras overnight. Nate
-
Hey guys I am getting out of discus and plants and getting into bigger cichlids Just wondering how many larger Americans I could keep in a 4x2x2 Was contemplating things like Green terrors Jaguar cichlids Synspilum Or flowerhorns I think the flowerhorns would have to go on their own if I got them Would it be better to only have one of each as they may get territorial if they pair up? S unsure as I have only ever kept discus I would just like a tank with some decent size fish in it .. Any suggestions would be great
-
Stocking a 5x2x2...
RJHELL posted a topic in General Aquarium DiscussionOkay guys help. Got a 5x2x2 and not real sure on what to fill it with. Was thinking of a mass rainbow tank but still not sure. Its a outside tank and gets plenty of sun in that area. So their WIlL be a pleco. Throw me your ideas. And i want more then just one fish so no jacks or togas or anything along them lines. Cheers RJ
-
Stocking a 60cm*30cm*27cm
BJplusMilo21 posted a topic in General Aquarium DiscussionHere is a planned stocking list for my 60cm*30cm*27cm aquarium: 4x Leopard Cory (but not until the tank has been established or around 2 months) 6x Celestial Pearl Danio 1x Steel Blue Killifish or 5x clown Killifish. Is this stocking list suitable for my aquarium? Note: If you have to replace the killifish please suggested another top dwelling species. Because they are being used as dither for the Pearl Dainio.
-
Hi all, First of all hope I'm in the right place here, am more than happy to be shifted if the mods think it better suited in another thread. As the title suggests, I've been stalking various fish forums for sometime now. Have picked up a couple of goods buys through this one's trading section, and have done quite a deal of reading via this and others. I now have a 6x2x2 set up with an FX5 and a 300W jager heater. Its currently pumping away in the lounge, cycling with 4 comet goldfish hopefully 'filthing' up the system and developing the necessary bacteria for me. Been on the go with fish for about 2 weeks now. The filter came second hand, and was only off line for about 12 or so hours, so I'm hoping I've managed to keep some culture working there. I've found what I think is a nice piece of wood, cleaned it up and soaked it for 2 weeks, and its now in with some pool filter sand and river rocks. I'm hoping to set up a SA river system of sorts. Don't have any planting yet, as the stock list will probably dictate whether that will be a waste of time anyway. With that in mind, I've now come to stocking and that's my question here for now. I have an older tank (picked up from a relative years ago, about 3") and it currently has 3 x 'Orange heads' geos (the largest, male I think, about 3", others about 2", have seen two batches of eggs from a pair. gobbled up in time by tankmates), 2 x native rainbows (each about 3"), 2 x clown loaches (about the 3" mark) and 3 x black tetras. Pretty sure the loaches have dined on some cavier here! My intention is to take the Geos out and put them into the 6"er, along with some mates. My current thoughts are: 3 x Orange head Geos (existing) + another 3 Some Jurapari Geos 4 or so Severum Blue Acara Uaru some dithers (tetras?) How does this sound? Am I overstocking? Am I mixing species that won't stand each other (I understand that egg laying can provide a whole new set of issues here.)? Any suggestions/advice are/is welcome. Thanks for the read, Cheers, Geoff.
-
Hi all, I've just had a 7 x 2 x 2 tank built and I'd like to set up a planted tank. At the moment I'm thinking 6 x discus 4 x neon blue rams 4 x gold rams 1 x flagtail 20 x rummynose 20 x cardinals 6 x whiptail catfish 8 x corydoras What do you think? I'm not 100% sure about keeping discus either so welcome to different stocking ideas! | https://www.qldaf.com/tags/stocking/ |
New duvet covers can be expensive, but they are relatively easy to make yourself. All that's required to make one are a decent amount of fabric and a few hours of cutting and sewing. These instructions are for a twin size duvet cover but the same information can be applied to make a queen or king duvet cover as well -- you'll just need more fabric.
Measure the duvet that you would like to cover. Most fabrics don't come in wide enough widths for a duvet so you will most likely need to panel your fabric vertically to achieve the correct width. The cover in this post is for a twin size duvet which measures 64" x 86". Adding for seam allowance, the total dimensions of your cut panels, when stitched together, should be 65" x 90 1/2".
There are many different options for paneling fabric -- you can choose two different fabrics or create panels from the same fabric, it's up to you.
Since the shape of your duvet cover is a rectangle, there is no need to create a paper pattern to cut around. Simply, mark off the length and width you'd like to cut with a ruler and pencil.
For the wider panels, you can fold the fabric in half and cut through two layers at once. For the narrower panels, line up the edge of your ruler with the edge of the fabric and mark off the width you'd like. Keep making marks along the length of the fabric as this will ensure that you will get a straight line to cut on.
Stitch together your vertical panels using a French seam. A french seam encloses all the seam allowance so your fabric does not fray when put into the washing machine. It also makes the inside of your cover very clean looking.
To create a french seam, pin the WRONG sides of your fabric together and stitch using a 1/4" seam allowance. Once you've made your seam, trim off about 1/8" of the seam allowance width.
Iron your seam to one side, flip the fabric over (RIGHT sides will be facing each other now) and fold at the seam line. Pin the seam closed and stitch on your machine with 1/4" seam allowance. After sewing, iron your seams flat to relax the stitching.
You should now have two duvet pieces, a top and a bottom. The next step is to stitch the hem of each piece. Fold up a 1/2" hem and iron flat. Then fold up 2" and iron again. Pin up the fold and stitch with your machine about 1/8" in from the fold line.
Now it is time to add buttonholes to one side of your duvet hem. You can use 4 buttons for this duvet, evenly spaced in the center of your duvet. First, measure the buttons you purchased so you now how big to make the holes. These buttons are 9/16" so the button holes will be 3/4" long. Center the buttonholes between the bottom edge and the stitch line. Since the hem is 2" wide and the buttonholes are 3/4" long, each buttonhole will be placed at 5/8" above the bottom of the fabric.
Use the buttonhole function on your sewing machine to stitch your buttonholes. Once all of the buttonholes are sewn, you need to use a seam ripper to cut them open. Start at the bottom of the buttonhole (above the thread), stick your seam ripper in and slide it to the top of the buttonhole.
Once your buttonholes are in, you can stitch the two sides of your duvet together. You can also use the French seam technique but if your fabric is heavy, it might get too thick at the corners. In that case, pin the right sides of your fabric together and stitch around the cover, starting from the left side of your buttonholes and ending on the right side. Then you can clean finish your seam allowance by using a zigzag stitch along the edge.
Flip your cover right side out and lie flat. Use your buttonholes to mark the correct placement of your buttons and stitch your buttons on. Place your duvet into your new cover and you're done!
Tip
-
If your duvet has loops of twill tape at the corners, you can stitch twill tape ties into the corners of your duvet to hold the duvet in place.
-
There are many different options to close your duvet cover besides buttonholes. You can use snaps, twill tape ties, even velcro!
-
The same instructions apply to create a queen or king size duvet as well, just make sure to measure your duvet to adjust for the amount of fabric needed. | https://ourpastimes.com/how-to-make-a-duvet-cover-12632359.html |
JLT Mobile Computers report development of sales volume for the first quarter 2020 and financial position as of 31 March
Växjö, Sweden, 7 April 2020 * * * The Board of Directors of JLT Mobile Computers, leading supplier of rugged computers for challenging environments, has resolved to disclose preliminary information about volume of sales and financial position for the first quarter 2020 due to uncertainties surrounding the COVID-19 situation.
The preliminary order intake for the first quarter amounts to MSEK 29 (31) and invoicing to MSEK 27 (37). Outgoing order backlog amounts to MSEK 14 (18).
The global market decline in financial instruments has impacted JLT’s investments, but it is estimated that this is offset by the strengthening of USD denominated holdings against the SEK. The total net liquid position amounts to MSEK 50, of which MSEK 19 is under discretionary management with a maximum of 20 percent in shares or share-based investments. The rest comprises bank balances, mainly in SEK and USD. The absolute majority of assets is cash. There are no interest-bearing liabilities.
The net liquid position of MSEK 50 is basically equivalent to the total operating costs of the company excluding cost of goods sold for 2019.
This information is information that JLT Mobile Computers AB (pub) is obliged to make public pursuant to the EU Market Abuse Regulation and the Securities Markets Act. The information was submitted for publication, through the agency of the contact person set out above, at 08:05 CET on 7 April 2020.
Want to learn more?
We’re here to help and advise you on every aspect of rugged devices and industry data communications. Please fill in the form and we will get back to you as soon as possible. | https://jltmobile.com/press_releases/jlt-mobile-computers-announces-sales-volume-for-q1-2020-and-financial-position-as-of-march-31st-swedish-only/ |
- This is the second episode that the Invisibility Ring appear in since its debut in Invisible Jake.
- The Pirate Magician's Lair makes a reappearance since Invisible Jake.
- This episode marks the first appearance of Beardini the Pirate Magician in the series.
- Beardini name is allusion of Harry Houdini.
- While searching the Pirate Magician's Lair Hook and Smee come across a wizard hat resembling Sorcerer Mickey's hat from Disney's Fantasia.
- The episode was directed by Broni Likomanov.
- The episode was written by Mark Drop.
- The storyboard was by Naz Ghodrati-azadi.
Community content is available under CC-BY-SA unless otherwise noted. | https://jakeandtheneverlandpirates.fandom.com/wiki/The_Remarkable_Beardini!/Trivia |
Enjoy an outdoor concert of chamber music by Haydn and Mozart performed by members of the H+H Orchestra at The Fells in Newbury, NH. Tickets for this concert are sold through The Fells' website. This concert will be on the Veranda unless the weather is inclement. We encourage you to bring your own picnic to enjoy during the concert. There will be limited tables and chairs for the event and so we encourage your to bring your own blanket and lawn chair. As The Fells' staff and and the performers need time to set up, we kindly ask that you do not arrive for a concert before 4:30pm. We appreciate your understanding.
Date
Sunday, August 29, 2021 at 5:00PM
Please note: Tickets are sold on The Fells' website here or by calling 603-763-4789 x3. | https://handelandhaydn.org/summer-concert-at-the-fells/ |
Job Description The primary function of an L2 Analyst is to ensure that the SOC team is performing its functions as required and to trouble shoot problematic incidents and events. In summary, the L2 Analyst shall also act as the technical SME and shall report technically to the L3 Analyst.
Responsibilities
• Work collaboratively with Account Manager for Client relations
• Track incident detection and closure
• Execute risk hunting activities
• Undertake forensic investigations
• Act as subject matter expert and expert witness where required
• General intelligence advisories and delegate intelligence aggregation tasks to L2
• Generate new use cases for emerging threats
• Conduct incident response coordination with customer
• Validation of security incidents
• Conduct audits of logging and correlation
• Conduct monthly security use case review and correlation audits
• Use of sandbox, honeypot, analytics tools and security testing
• Escalation management
• Ensure process compliance
• Ensure quality of investigations and notification and direct L2 and L1 accordingly
• Report deviations to SOC manager and L3
• Ensure SLA compliance for projects within remit
• Perform deep analysis to security incidents to identify the full kill chain
• Set up weekly meeting to review the weekly reports with the client
• Respond to clients’ requests, concerns and suggestions
• Act as subject matter expert for different clients
• Provide knowledge to L1 and L2 such as guides, cheat sheets etc
• Follow up with the recommendations to the client to contain an incident or mitigate a threat
• Conduct presentations and updates to the client
• Respond to incident escalations and provide solid recommendations
• Update aging incidents and requests
• Track SOC performance in terms of SLAs and incident quality
• Review vulnerability assessment reports with the client and provide necessary recommendations
• Configure and maintain vulnerability scanners policies and reports
• Conduct threat hunting exercises on SIEM and EDR platforms
• Conduct penetration testing on web applications, mobile applications, servers (Windows/Linux) and wireless infrastructure
• Develop and improve processes for monitoring and incident qualification
• Perform quarterly evaluation for L1 and L2 analysts and report feedback to SI management
• Participate in professional services (internal and external penetration testing, wireless assessments, web and mobile application assessments, firewall and server security audits, social engineering exercises, security awareness programs etc.)
• Perform threat intelligence analysis and investigations. Search on the dark web and use other platforms such as RF to identify intelligence indicators or threats for a specific client
• Create reports for threat intelligence as a service. | https://www.securityhq.com/careers/cyber-security-analyst-l2/ |
CROSS REFERENCE TO RELATED APPLICATION
BACKGROUND
SUMMARY
EMBODIMENT
1. Overall Configuration
2. White Keys
3. Black Keys
4. Frame
11
a
4.1. Support Frame Portion
11
b
4.2. Support Frame Portion
11
c
4.3. Support Frame Portion
5. Keys and Connectors
5. 1. White Keys
5.1.1. Front Narrow Portion
5.1.2. Wide Portions
5.2. Black Keys
5.2.1. Back Narrow Portion
5.3. Attachment and Detachment of Connector
6. Arrangement in Connector
6.1. Pivot Center
6.2. Coupler
6.3. First Modification
6.4. Second Modification
7. Hammer Mechanism
The present application is a continuation application of International Application No. PCT/JP2017/009165, filed on Mar. 8, 2017, which claims priority to Japanese Patent Application No. 2016-061657, filed on Mar. 25, 2016. The contents of these applications are incorporated herein by in their entirety.
The present disclosure relates to techniques for a keyboard apparatus and an electronic keyboard instrument using the keyboard apparatus.
Patent Document 1 (Japanese Patent Application Publication No. 2008-191650) discloses a technique relating to a keyboard apparatus including: a key; a horizontal hinge portion connected from the key in a key-longitudinal back direction; and a vertical hinge portion connected from the horizontal hinge portion in the key-longitudinal back direction.
In this technique, in the case where the key is coupled to the frame, the coupler requires a large width in a scale direction, leading to a possibility of contact between the couplers when the key is moved in the scale direction.
It is therefore an object of the present disclosure to provide a keyboard apparatus configured to reduce contact between couplers in a case where a configuration in which keys are coupled to a frame is employed.
An aspect of the present disclosure relates to a keyboard apparatus includes: a plurality of keys; at least one frame configured to support at least one of the keys; at least one bendable portion disposed between one of the keys and the frame and having flexibility in a scale direction; and a coupler configured to couple the bendable portion and the key to each other attachably and detachably, wherein two couplers each as the coupler which correspond respectively to the keys adjacent to each other are disposed respectively at positions different from each other in a longitudinal direction of the key.
Another aspect of the present disclosure relates to an electronic keyboard instrument including: a keyboard apparatus including (i) a plurality of keys, (ii) at least one frame configured to support at least one of the keys, (iii) at least one bendable portion disposed between one of the keys and the frame and having flexibility in a scale direction, and (iv) a coupler configured to couple the bendable portion and the key to each other attachably and detachably, wherein two couplers each as the coupler which correspond respectively to the keys adjacent to each other are disposed respectively at positions different from each other in a key longitudinal direction of the key; a sensor configured to detect operation for the key; and a sound source section configured to produce a sound waveform signal in response to a signal output by the sensor.
500
Hereinafter, there will be described an electronic keyboard instrument according to one embodiment of the present disclosure by reference to the drawings. It is to be understood that the following embodiment is one example of the embodiment of the present disclosure, and the present disclosure is not limited to the embodiment.
FIG. 1
FIG. 1
500
100
500
501
100
51
51
502
503
is a perspective view of the electronic keyboard instrument including a keyboard apparatus according to one embodiment of the present disclosure. As illustrated in , the electronic keyboard instrument includes: a housing ; the keyboard apparatus including white keys W and black keys B; a cover ; and a cover .
100
501
502
501
502
100
503
501
100
100
100
503
100
503
FIG. 2
The keyboard apparatus is installed on the housing . The cover is openable and closable with respect to the housing . When being in a closed state, the cover covers the entire keyboard apparatus . The cover is immovably secured to the housing and is configured to cover a portion of the keyboard apparatus . The keyboard apparatus includes: a visible portion X not to be covered with the cover ; and a non-visible portion Y (see ) to be covered with the cover .
FIG. 2
500
100
1
2
is an enlarged view of a portion of the electronic keyboard instrument . In the following explanation, a direction directed from the near side toward the far side for a user in a key longitudinal direction M in the keyboard apparatus will be referred to as “key-longitudinal back direction M”, and a direction directed from the far side toward the near side for the user in the key longitudinal direction M will be referred to as “key-longitudinal front direction M”.
100
51
51
51
52
52
52
60
51
52
51
1
51
60
51
52
In the keyboard apparatus , the keys (the white keys W and the black keys B), connectors (white-key connectors W and black-key connectors B), and a frame are arranged in this order in the key longitudinal direction M. Each of the keys is a component to be depressed by the user. Each of the connectors extends from a corresponding one of the keys in the key-longitudinal back direction M and is connected between the corresponding key and the frame . A plurality of sets of the keys and the connectors respectively coupled to each other are arranged in a scale direction S.
60
1
52
60
60
60
60
60
60
60
60
60
60
a
a
a
The frame is disposed on a key-longitudinal-back-direction-M side of the connectors in the key longitudinal direction M. The frame includes a supporter , a plurality of frame narrow portions W, and a plurality of frame narrow portions B. The supporter extends in the scale direction S and supports the frame narrow portions W and the frame narrow portions B. Each of the frame narrow portions W and the frame narrow portions B extends from the supporter in a direction substantially orthogonal to the scale direction S (the key longitudinal direction M).
51
100
100
52
51
100
100
503
FIG. 1
FIG. 1
Portions of the keys which correspond to the visible portion X of the keyboard apparatus are disposed at a region viewable from the outside (also see ). The connectors and the other portions of the keys which correspond to the non-visible portion Y of the keyboard apparatus are disposed at a region covered with the cover and not viewable from the outside (also see ).
51
51
21
51
22
51
21
51
22
51
21
51
22
The white keys W include a white key W (a first key) and a white key W (a second key) having the same shape. For example, the white key W (the first key) and the white key W (the second key) are different from each other by one octave and have the same shape. Thus, since the white key W, W have the same shape, one white key may also be used for another white key corresponding to another octave.
51
51
1
51
2
51
1
51
2
51
51
The black keys B include a black key B (the first key) and a black key B (the second key) having the same shape. For example, the black key B (the first key) and the black key B (the second key) are arranged with one or two white keys W interposed therebetween and have the same shape. Thus, since the black keys B have the same shape, the black key may be used any position of the black key.
FIG. 3
FIG. 3
100
51
11
11
11
11
11
11
11
11
11
a
b
c
b
c
a
a
c
is a side view of the keyboard apparatus , with the white key W being viewed from a lateral side. As illustrated in , a frame includes a support frame portion , a support frame portion , and a support frame portion . The support frame portion and the support frame portion are secured to the support frame portion . The support frame portions -are connected to each other immovably relative to each other.
11
11
12
12
12
11
12
55
51
12
12
2
12
12
1
a
b
a
FIG. 3
FIG. 8
The support frame portion includes a pivot shaft X and supports a hammer such that the hammer is pivotable. The hammer pivots about the pivot shaft X (indicated by the broken line in ). The hammer is configured such that, when a pressing portion extending downward from the white key W is moved downward, a basal end portion (see ) of the hammer on a key-longitudinal-front-direction-M-side is moved downward, and a distal end portion of the hammer on the key-longitudinal-back-direction-M side is moved upward.
11
14
14
12
12
51
14
12
12
1
11
12
2
11
12
12
11
14
12
12
b
a
a
a
The support frame portion supports a supporter . The supporter stops and supports, from below, the distal end portion of the hammer moving downward by gravity when the white key W is in a non-depressed state. The supporter extends in the scale direction S. The hammer is set such that a portion of the hammer on the key-longitudinal-back-direction-M side of the pivot shaft X is longer than a portion of the hammer on the key-longitudinal-front-direction-M-side of the pivot shaft X. Accordingly, in the non-depressed state, the distal end portion of the hammer is located on a lower side of the pivot shaft X due to gravity. The supporter defines a lower limit of a range of pivotal movement of the distal end portion of the hammer .
11
13
13
12
12
51
13
11
c
a
c
FIG. 8B
The support frame portion supports a hammer stopper . The hammer stopper contacts the distal end portion of the hammer which moves upward when the white key W is in the depressed state (). Each of the hammer stopper and the support frame portion extends in the scale direction S.
FIG. 4A
FIG. 4B
FIG. 4C
FIG. 5A
FIG. 5B
51
51
70
60
51
51
52
51
6
7
70
60
60
52
51
6
7
8
60
60
51
is a plan view of the white key W, and is a side view of the white key W. is a side view of a portion of a configuration of one of couplers and a corresponding one of the frame narrow portions W before their coupling. is a plan view of the black key B, and is a side view of the black key B. Each of the white-key connectors W connected to the respective white keys W includes a front narrow portion (a second region), a wide portion (a first region), and the coupler . The frame includes the frame narrow portions W. Likewise, each of the black-key connectors B connected to the respective black keys B includes the front narrow portion (the second region), the wide portion (the first region), and a back narrow portion (the second region). The frame includes the frame narrow portions B. The following description is provided, focusing on the white keys W.
60
60
60
60
60
60
60
60
60
60
51
52
60
b
a
d
d
b
c
b
d
d
FIGS. 4B and 5B
Each of the frame narrow portions W includes: a bendable portion (the second region) extending from the supporter and flexible in the scale direction S; and a bendable portion (a hinge) flexible in both of the scale direction S and a vertical direction E. Here, a portion of the frame narrow portion W which is different from the bendable portion corresponds to the bendable portion , and a cutout portion corresponds to a portion of edges of the bendable portion and the bendable portion . The key and the connector are pivotable from a portion of the bendable portion in the vertical direction E (see ).
60
70
60
60
70
70
60
60
70
70
60
70
51
FIG. 4C
FIG. 4C
e
e
f
f
It is noted that the frame narrow portion W and the coupler illustrated in are coupled to each other. For this coupling, a first inserted portion of the frame narrow portion W is inserted in an insertion opening of the coupler , and a second inserted portion of the frame narrow portion W is inserted in an insertion opening of the coupler . The attaching and detaching mechanism in is also applied to an attaching and detaching mechanism of the frame narrow portion B and the coupler for the black key B.
60
60
60
60
51
60
51
100
b
b
d
d
Since the bendable portion has a flat surface extending in the direction substantially orthogonal to the scale direction S, the bendable portion is flexible in the scale direction S and bendable in the scale direction S. The bendable portion is flexible in the scale direction S and bendable in the scale direction S and also is flexible in the vertical direction E and bendable in the vertical direction E. Accordingly, it is possible to consider that the bendable portion is flexible in the pivotal direction of the key . Thus, the frame provides a pivotal-movement function of the key . As a result, the configuration of the keyboard apparatus is simplified.
6
51
1
2
6
4
51
2
6
2
6
6
The front narrow portion (also referred to as “second region”, “first narrow portion”, or “first low-stiffness portion”) extends from the white key W in the key-longitudinal back direction M. The width S of the front narrow portion in the scale direction S is less than the width S of the white key W in the scale direction S. The width S of the front narrow portion in the scale direction S is less than the thickness H of the front narrow portion in the vertical direction E. Briefly, the front narrow portion is disposed such that a thin plate-like member is oriented vertically.
2
6
6
51
6
6
51
51
Thus, since the width S of the front narrow portion in the scale direction S is small, the stiffness of the front narrow portion in the scale direction S is less than that of the white key W in the scale direction S, and accordingly the front narrow portion is flexible in the scale direction S and in a yawing direction Y and easily bendable. The configuration of the front narrow portion is the same in the case of the black keys B and in the case of the white keys W.
7
1
6
51
1
7
2
6
The wide portion (also referred to as “first region” or “high-stiffness portion”) extends in the key-longitudinal back direction M from the front narrow portion located near the white key W. The width S of the wide portion in the scale direction S is greater than the width S of the front narrow portion in the scale direction S.
51
7
7
6
51
7
4
51
Since the width of the wide portion in the scale direction S is large, the stiffness of the wide portion in the scale direction S is greater than that of the front narrow portion in the scale direction S. It is noted that the width of the wide portion in the scale direction S is less than the width S of the key in the scale direction S.
7
7
7
7
7
1
6
7
6
60
7
a
a
a
a
a.
The wide portion has a recessed portion that is recessed upward in side view. Though the recessed portion reduces the stiffness of the wide portion , the recessed portion has the width S greater than the width of the front narrow portion , and accordingly the recessed portion has high stiffness. The front narrow portion and the frame narrow portions W only need to be formed in at least a portion of a region different from the recessed portion
1
7
1
7
7
7
1
7
2
6
a
It is noted that the width S of the wide portion in the scale direction S is less than the thickness H, in the vertical direction E, of a thin portion of the wide portion due to the recessed portion formed therein. Briefly, the wide portion is disposed such that a thin plate-like member is oriented vertically. The thickness H of the wide portion in the vertical direction E is less than the thickness H of the front narrow portion in the vertical direction E.
51
FIGS. 5A and 5B
There will be next described the black keys B with reference to .
7
51
51
1
7
52
51
1
7
51
51
51
52
52
1
1
7
2
2
6
The configuration of the wide portion is similar in the case of the black keys B and in the case of the white keys W. However, the length n, in the key longitudinal direction M, of the wide portion of the black-key connector B connected to the black key B is less than the length N of the wide portion of the white key W in the key longitudinal direction M. This is partly because the black key B is less than the white key W in length in the key longitudinal direction M. Independently of the black-key connectors B and the white-key connectors W, each of the lengths n, N of the wide portion in the key longitudinal direction M is greater than a corresponding one of the lengths n, N of the front narrow portion in the key longitudinal direction M.
8
52
8
7
1
3
8
1
7
4
51
3
8
3
8
8
There will be next described the back narrow portions of the respective black-key connectors B. The back narrow portion (also referred to as “second region”, “second narrow portion”, or “second low-stiffness portion”) extends from the wide portion in the key-longitudinal back direction M. The width S of the back narrow portion in the scale direction S is less than each of the width S of the wide portion in the scale direction S and the width S of the key B in the scale direction S. The width S of the back narrow portion in the scale direction S is less than the thickness H of the back narrow portion in the vertical direction E. Briefly, the back narrow portion is disposed such that a thin plate-like member is oriented vertically.
8
3
8
7
8
Thus, it is possible to consider that the back narrow portion has a shape in which, since its width S in the scale direction S is small, the stiffness of the back narrow portion in the scale direction S is less than that of the wide portion in the scale direction S, and accordingly the back narrow portion is flexible in the scale direction S and in the yawing direction Y and easily bendable in the scale direction S.
3
8
2
6
3
8
2
6
In the present embodiment, the width S of the back narrow portion in the scale direction S is substantially equal to the width S of the front narrow portion in the scale direction S. However, the width S of the back narrow portion in the scale direction S may be greater or less than the width S of the front narrow portion in the scale direction S.
2
6
1
7
6
7
6
1
7
3
8
5
60
60
7
8
1
7
3
8
As described above, the width S of the front narrow portion in the scale direction S may be less than the width S of the wide portion in the scale direction S. Accordingly, the stiffness of the front narrow portion in the scale direction S is less than that of the wide portion in the scale direction S, and accordingly the front narrow portion is flexible in the scale direction S and in the yawing direction Y and easily bendable. The width S of the wide portion in the scale direction S is greater than each of the width S of the back narrow portion in the scale direction S and the width S of each of the frame narrow portions W, B in the scale direction S. Accordingly, the stiffness of the wide portion in the scale direction S is greater than that of the back narrow portion in the scale direction S. The thickness H of the wide portion in the vertical direction E is less than the thickness H of the back narrow portion in the vertical direction E.
51
6
2
7
60
1
7
51
52
60
6
60
6
60
FIG. 2
In the case of the white key W in the present embodiment, the front narrow portion is disposed on the key-longitudinal-front-direction-M-side (the near side) of the wide portion , and the frame narrow portions W is disposed on the key-longitudinal-back-direction-M side (the far side) of the wide portion . When the white key W is deformed in the yawing direction Y, a positional relationship between the connector (see ) and the frame changes. The front narrow portion and the frame narrow portion W have a function of reducing the effects of the change in the positional relationship by deformation of the front narrow portion and the frame narrow portion W.
51
6
2
7
8
60
1
7
51
52
60
6
8
60
6
8
60
FIG. 2
In the case of the black key B in the present embodiment, the front narrow portion is disposed on the key-longitudinal-front-direction-M-side (the near side) of the wide portion , and the back narrow portion and the frame narrow portions B are disposed on the key-longitudinal-back-direction-M side (the far side) of the wide portion . When the black key B is deformed in the yawing direction Y, a positional relationship between the connector (see ) and the frame changes. The front narrow portion , the back narrow portion , and the frame narrow portions B have a function of reducing the effects of the change in the positional relationship by deformation of the front narrow portion , the back narrow portion , and the frame narrow portions B.
51
70
7
60
51
70
8
60
51
51
60
8
60
As described above, in the case of the white key W, the coupler is disposed between the wide portion and the frame narrow portion W in the key longitudinal direction M. In the case of the black key B, the coupler is disposed between the back narrow portion and the frame narrow portion B in the key longitudinal direction M. There is such a difference between the white key W and the black key B. However, the length of the frame narrow portion W in the key longitudinal direction M is substantially equal to the sum of the lengths of the back narrow portion and the frame narrow portions B in the key longitudinal direction M.
51
6
7
70
60
6
7
70
70
7
60
In the case of the white key W, the front narrow portion , the wide portion , the coupler , and the frame narrow portion W are arranged in this order in the key longitudinal direction M. The front narrow portion , the wide portion , and the couplers are formed as a unit. The coupler formed integrally with the wide portion is coupled to the frame narrow portion W attachably and dettachably.
51
6
7
8
70
60
6
7
8
70
70
8
60
8
2
70
60
1
70
In the case of the black key B, the front narrow portion , the wide portion , the back narrow portion , the coupler , and the frame narrow portion B are arranged in this order in the key longitudinal direction M. The front narrow portion , the wide portion , the back narrow portion , and the coupler are formed as a unit. The coupler formed integrally with the back narrow portion is coupled to the frame narrow portion B attachably and dettachably. The back narrow portion is located on the key-longitudinal-front-direction-M-side of the coupler , and the frame narrow portion B is located on the key-longitudinal-back-direction-M side of the coupler .
70
51
60
70
100
100
b
The coupler is attached and detached by its sliding movement with the key and the bendable portion in the up and down direction. This configuration enables the coupler to be attached and detached only by its sliding movement, resulting in improved workability in manufacture of the keyboard apparatus . Also, the mechanical strength is improved, thereby improving the durability of the keyboard apparatus against an external force produced when the key is depressed.
FIG. 6
FIG. 6
100
70
60
60
60
60
51
b
a
is an enlarged view of a portion of the keyboard apparatus , illustrating the configurations of the couplers , the bendable portions , and the frame . In , the frame is slightly inclined with respect to the scale direction S so as to extend in the lower right direction, but this illustration exaggerates the inclination for emphasis. That is, the distance between the supporter and the key is smaller on a high-pitched-sound side than on a low-pitched sound side.
FIG. 6
1
60
51
1
51
1
51
2
51
2
60
51
2
1
1
51
1
51
1
2
2
51
2
51
2
b
b
As illustrated in , the pivot center J of the bendable portion for a white key W corresponding to a low-pitched sound is located farther from the white keys W, W in the key longitudinal direction M of the white key W than the pivot center J of the bendable portion for a white key W corresponding to a high-pitched sound. That is, the distance h between the pivot center J of the white key W corresponding to the low-pitched sound and the white key W is greater than the distance h between the pivot center J of the white key W corresponding to the high-pitched sound and the white key W.
1
70
51
1
1
51
1
2
70
51
2
2
51
2
1
2
1
1
51
1
51
1
2
2
51
2
51
2
The distance k between a first coupler corresponding to the white key W and the pivot center J of the white key W is different from the distance k between a second coupler corresponding to the white key W and the pivot center J of the white key W. Thus, making the distance k and the distance k different from each other is only required to make the distance h between the pivot center J of the white key W and the white key W and the distance h between the pivot center J of the white key W and the white key W different from each other.
1
60
51
1
51
1
51
2
51
1
2
60
51
2
3
1
51
1
51
1
4
2
51
2
51
2
b
b
The pivot center B of the bendable portion for the black key B corresponding to a low-pitched sound is farther from the black keys B, B in the key longitudinal direction M of the black key B than the pivot center B of the bendable portion for the black key B corresponding to a high-pitched sound. That is, the distance h between the pivot center B of the black key B corresponding to the low-pitched sound and the black key B is greater than the distance h between the pivot center B of the black key B corresponding to the high-pitched sound and the black key B.
51
1
51
2
1
60
51
1
51
1
51
2
2
60
51
2
3
1
51
1
51
1
2
2
51
2
51
2
b
b
In the case of the black key B and the white key W arranged in a direction directed from the low-pitched sound toward the high-pitched sound, the pivot center B of the bendable portion for the black key B on the low-pitched-sound side is disposed farther from the black key B and the white key W in the key longitudinal direction M than the pivot center J of the bendable portion for the white key W on the high-pitched-sound side. That is, the distance h between the pivot center B of the black key B on the low-pitched-sound side and the black key B is greater than the distance h between the pivot center J of the white key W on the high-pitched-sound side and the white key W.
60
60
51
51
100
b
b
As described above, the pivot center of the bendable portion for the key corresponding to the low-pitched sound is located farther from the key in the key longitudinal direction M of the key than the pivot center of the bendable portion for the key corresponding to the high-pitched sound. Since the pivot center J of the key corresponding to the low-pitched sound is far from the key , it is possible to achieve a touch feeling similar to that in a ground piano. This improves operability of the keyboard apparatus .
70
51
1
70
51
1
70
51
1
70
51
1
70
51
1
70
51
1
The coupler for the white key W and the coupler for the black key B which are adjacent to each other are different from each other in position in the key longitudinal direction M. In other words, the coupler for the white key W and the coupler for the black key B which are adjacent to each other are respectively arranged at positions not overlapping each other in the scale direction S. This configuration reduces contact between the coupler for the white key W and the coupler for the black key B which are adjacent to each other.
70
51
1
60
51
1
70
51
1
60
51
1
70
51
1
51
1
b
b
The coupler for the black key B is opposed to the bendable portion for the white key W in the scale direction S. Thus, the coupler for the black key B and the bendable portion for the white key W are respectively arranged at positions spaced apart from each other at a predetermined distance, ensuring a large movable area of the coupler for the black key B. This results in improved stability of operation of the black key B. This applies to the other black keys.
70
51
1
8
51
1
8
51
1
70
8
The coupler for the white key W is opposed to the back narrow portion of the black key B in the scale direction S. Here, as described above, the back narrow portion (a connecting bendable portion) connects between the black key B (one of the plurality of keys) and the coupler . The back narrow portion has a flat surface extending in a direction substantially orthogonal to the scale direction S and has flexibility.
70
51
1
8
70
51
1
51
1
The coupler for the white key W and the back narrow portion are opposed to each other and respectively arranged at positions spaced apart from each other at a predetermined distance, ensuring a large movable area of the coupler for the white key W. This results in improved stability of operation of the white key W. This applies to the other white keys.
70
51
51
51
70
51
3
70
51
1
51
1
4
70
51
2
51
2
70
51
1
70
51
70
51
70
51
100
70
The coupler for the black key B is provided farther from the key in the key longitudinal direction M of the key than the coupler for the white key W. That is, the distance k between the coupler for the black key B and the black key B is greater than the distance k between the coupler for the white key W and the white key W. With this configuration, the coupler for the black key B can be disposed on the key-longitudinal-back-direction-M side of the coupler for the white keys W. As a result, the coupler for the black key B and the coupler for the white key W are not adjacent to each other in the scale direction S, resulting in reduction in breakage and deterioration of the keyboard apparatus due to contact between the couplers .
FIG. 7
FIG. 7
100
100
61
52
51
70
70
61
7
60
61
60
60
70
7
b
a
is an enlarged view of a portion of the keyboard apparatus according to a modification of the embodiment of the present disclosure. As illustrated in , the keyboard apparatus according to the present modification includes a plate portion between the connectors W of the respective white keys W. In this case, a surface L of the coupler which faces the plate portion may be flush with a side surface of the wide portion (or the bendable portion ). Here, the wording “flush with” means that the surfaces are located within the same plane in the scale direction S. Here, the plate portion is a portion of the frame which supports the supporter and extends in the key longitudinal direction M. It is noted that the couplers for the other keys protrudes in the scale direction S beyond the wide portion .
70
70
70
61
7
61
70
61
70
61
70
The coupler does not protrude in the scale direction S in order to make the distance between the surface L of the coupler and the plate portion in the scale direction S and the distance between the side surface of the wide portion and the plate portion in the scale direction S substantially equal to each other. Thus, the coupler opposed to the plate portion may be different in shape from the coupler not opposed to the plate portion among the plurality of couplers .
70
61
60
61
70
51
61
This configuration ensures a large distance between the coupler and the plate portion . With this configuration, even in the case of the frame having the plate portion , the coupler for the white key W moving in the vertical direction E does not contact the plate portion .
FIG. 10
FIG. 10
FIG. 2
100
60
60
60
1
60
6
60
1
60
6
51
60
51
60
1
60
6
60
51
60
60
51
80
60
60
a
a
a
a
a
b
a
a
a
b
a
a
b
is an enlarged view of a portion of the keyboard apparatus according to a modification. As illustrated in , the supporter of the frame may be divided into supporters -. Thus, each of the supporters - may support a single key . In this case, at least one bendable portion flexible in the scale direction S is disposed between each of the keys and a corresponding one of the supporters -. As illustrated in , the one supporter may be configured to support two or more keys such that a plurality of the bendable portions are disposed between the supporter and a plurality of keys . It is noted that a bolt may be used such that the supporter and the bendable portion are attachable to and detachable from each other.
FIG. 8A
FIG. 8B
51
12
51
51
12
51
30
12
51
13
12
30
11
14
c
is a side view illustrating a positional relationship between the white key W and the hammer when the white key W is in the non-depressed state. is a side view illustrating a positional relationship between the white key W and the hammer when the white key W is in the depressed state. A hammer mechanism includes: the hammer configured to be operated in response to depression of the white key W; and the hammer stopper configured to limit operation of the hammer . The hammer mechanism further includes the support frame portion and the supporter .
7
13
12
11
13
a
c
The above-described recessed portion is recessed so as to avoid the hammer stopper that contacts the hammer , and at least a portion of the support frame portion supporting the hammer stopper .
11
7
51
7
1
11
2
7
11
51
51
11
7
11
7
c
a
a
c
a
c
c
a
c
a
FIG. 8B
The support frame portion is disposed substantially parallel with the recessed portion in the depressed state of the white key W (see ). In this state, the most-recessed surface of the recessed portion indicated by an imaginary line Q and a surface of the support frame portion indicated by an imaginary line Q are close at the distance d. Thus, in the case where the recessed portion and the support frame portion are configured so as to be located as close as possible when the white key W is depressed, a space under the key can be used without any unnecessary portion, and an unnecessary space is reduced to the distance d. It is noted that the support frame portion and the recessed portion may not be parallel with each other as long as the support frame portion and the recessed portion are configured so as to be brought into closer to each other as possible.
12
12
2
11
81
51
12
81
85
81
c
c
The hammer includes a sensor pressing portion on the key-longitudinal-front-direction-M-side of the pivot shaft X. A sensor configured to detect depression (operation) of the key is disposed under the sensor pressing portion . There will be described the sensor and a sound-source device connected to the sensor .
FIG. 9
85
85
82
83
84
81
100
81
100
81
is a block diagram illustrating a configuration of the sound-source device . The sound-source device includes a signal converter section , a sound source section , and an output section . Sensors are provided corresponding to the respective keys . Each of the sensors detects an operation of a corresponding one of the keys and outputs signals in accordance with the detection. In the present example, each of the sensors outputs signals in accordance with three levels of key pressing amounts. The speed of the key pressing is detectable in accordance with a time interval between the signals.
82
81
81
1
81
2
81
88
100
100
82
100
100
82
82
The signal converter section obtains the signals output from the sensors (the sensors -, -, . . . , - corresponding to the respective 88 keys ) and creates and outputs an operation signal in accordance with an operation state of each of the keys . In the present example, the operation signal is a MIDI signal. Thus, the signal converter section outputs “Note-On” when a key is pressed. In this output, a key number indicating which one of the 88 keys is operated, and a velocity corresponding to the speed of the key pressing are also output in association with “Note-On”. When the player has released the key , the signal converter section outputs the key number and “Note-Off” in association with each other. A signal created in response to another operation, such as an operation on a pedal, may be output to the signal converter section and reflected on the operation signal.
83
82
84
83
80
The sound source section creates the sound waveform signal based on the operation signal output from the signal converter section . The output section outputs the sound waveform signal created by the sound source section . This sound waveform signal is output to the speaker or a sound-waveform-signal output terminal, not illustrated, for example.
FIG. 8
30
7
7
51
30
7
30
7
1
2
7
a
a
a
a
Returning to the explanation for , a portion of the hammer mechanism is located at the recessed portion of the wide portion in the state in which the white key W is depressed as described above. The wordings “a portion of the hammer mechanism is located at the recessed portion ” mean that a portion of the hammer mechanism is located within a region enclosed by the recessed portion and an imaginary line P connecting between one end portion P and the other end portion P of the recessed portion in the key longitudinal direction M.
13
7
7
51
13
7
7
51
51
13
7
7
13
7
7
51
a
a
a
a
FIG. 8B
FIG. 8A
In the present embodiment, specifically, the hammer stopper is disposed so as to be located at the recessed portion of the wide portion in the state in which the key is depressed (see ). In the present embodiment, the hammer stopper is disposed so as to be located at the recessed portion of the wide portion even in the state in which the key is not depressed (see ). It is noted that, in the case where the key is not depressed, even when the hammer stopper is located at the recessed portion of the wide portion , the hammer stopper may be positioned at the recessed portion of the wide portion after the key is depressed.
13
12
12
51
12
13
51
51
a
The hammer stopper has a function of limiting an upper-limit position of the distal end portion of the hammer when the white key W is depressed by the player. Since the hammer is brought into contact with the hammer stopper , when the white key W is depressed by the player, the player is given a feeling of depression of the key as in a ground piano.
70
51
70
70
51
51
In the configuration in the present embodiment, the couplers (the connectors) are provided at different positions in the key longitudinal direction M of the key , whereby the couplers are not adjacent to each other in the scale direction S. This configuration reduces contact between the couplers in the case where the key is moved in the scale direction S. This results in improved stability of operation of the key .
7
52
51
51
12
30
7
7
52
51
51
13
12
12
500
12
51
a
a
a
a
In the above-described configuration in the present embodiment, the recessed portion is formed in the connector . When the white key W or the black key B is depressed, the hammer of the hammer mechanism can be moved toward the recessed portion . The recessed portion formed in the connector for the white key W or the black key B can be used as a space for receiving the hammer stopper and a space into which the distal end portion of the hammer is moved. This configuration reduces the size of the electronic keyboard instrument in the vertical direction E and ensures a large movable range of the hammer . Also, a heavy touch of the key is achieved.
6
8
52
In the configuration in the present embodiment, the flexibility of the front narrow portion and the back narrow portion is maintained, and the stiffness of the connector in the vertical direction E is maintained.
Configurations with constituent elements added, deleted, or changed in design or with steps added, omitted, or changed in condition by those skilled in the art based on the configuration explained as the embodiment of the present disclosure are also included in the scope of the present disclosure as long as the configurations contain the spirit of the present disclosure.
Even in the case where effects different from the effects achieved by the above-described embodiment are achieved, when the effects are obvious from the description of the present specification or easily predictable by those skilled in the art, the effects are interpreted as being achieved by the present disclosure.
BRIEF DESCRIPTION OF THE DRAWINGS
The objects, features, advantages, and technical and industrial significance of the present disclosure will be better understood by reading the following detailed description of the embodiments, when considered in connection with the accompanying drawings, in which:
FIG. 1
is a perspective view of an electronic keyboard instrument including a keyboard apparatus according to one embodiment of the present disclosure;
FIG. 2
is an enlarged view of a portion of the electronic keyboard instrument;
FIG. 3
is a side view of the keyboard apparatus;
FIG. 4A
FIG. 4B
FIG. 4C
70
60
is a plan view of a white key, is a side view of the white key, and is a side view of a portion of a configuration of one of couplers and a corresponding one of the frame narrow portions W before their coupling;
FIG. 5A
FIG. 5B
is a plan view of a black key, and is a side view of the black key;
FIG. 6
is an enlarged view of a portion of the keyboard apparatus;
FIG. 7
is an enlarged view of a portion of a keyboard apparatus according to a modification of the embodiment of the present disclosure;
FIG. 8A
FIG. 8B
is a side view indicating a positional relationship between the white key and a hammer when the white key is in a non-depressed state, and is a side view indicating a positional relationship between the white key and the hammer when the white key is in a depressed state;
FIG. 9
is a block diagram illustrating a configuration of a sound-source device; and
FIG. 10
is an enlarged view of a portion of a keyboard apparatus according to a modification. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.