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For those that haven't seen this link - please have a look as it's a an awe-inspiring 360 interactive panormic image of the milky way (may need a bit of time to load though!). I've found myself idly browsing it several times and as an astronomy layman I wondered if anyone could help me navigate what I'm looking at.
While I can orientate myself and visualise where we are in the milky way, what is the distinctive black band? I'm assuming it is caused by the absence of stars in these areas. However, as I understood it, the disc (as well as the center) are the areas of highest concentration of stars. Are these 'gaps' caused by dark matter?
The black band is not absence of stars, but rather clouds of gas and dust — a significant component of almost all spiral galaxies$^\dagger$ — which block the light of background stars and luminous gas.
In the image, you see both individual stars, scattered all over, and the distinctive bright band of the Milky Way, most of the stars of which are so far away that they blend together. Most of the dust is in the plane of the Milky Way. The individual stars are closer to us than the bright band and the dust clouds, so what you see is the billions of stars in the Milky Way band, some of the light of which is blocked by dust, and then on top of this you see a few 1000s individual stars that are closer.
Galaxies consist of roughly 85% dark matter and 15% normal ("baryonic") matter. By far, most of the normal matter is hydrogen and helium, some of which is locked up in stars, and some of it in huge gas clouds, sometimes glowing (the pink clouds you see in the Milky Way image are probably hydrogen clouds being excited by the hard UV radiation from hot and massive stars, subsequently emitting H$\alpha$ light). A small fraction (1–2%) of the normal matter are heavier elements, by astronomers lazily referred to as "metals". Roughly 2/3 of the metals are in the gas phase, but the remaining 1/3 has depleted into dust grains, e.g. silicates and soot. This dust is mixed with the gas clouds and often become dense enough that they block light of stars.
Dark matter cannot be seen. It's… well, dark. It interacts with normal matter and light only through gravity. That means that if you could place a lump of dark matter in front of a star (you can't, really), you wouldn't block its light. It would pass right through. If the lump were big enough, it might gravitationally deflect the light, so the background star would look distorted to you, as would a lump of normal matter (e.g. a black hole).
$^\dagger$In contrast, the interstellar medium of elliptical galaxies tend to be much more gas- and dust-depleted.
Pela has a nice answer, I'll just add my 2 cents.
While I can orientate myself and visualise where we are in the milky way, what is the distinctive black band? I'm assuming it is caused by the absence of stars in these areas.
The black band probably blocks some of the more dense areas of stars, not a lack of stars at all, with the exception, I think, for the larger dark area on the far left, which might be a lack of Milky-way stars. I believe that window of low star density is where Hubble spends most of it's time looking into deep space.
Also, consider which is brighter a flashlight 6 inches from your eye or a car head-light 60 feet away. The center of our galaxy may be a lot brighter, but we're pretty far away from it, so the relative brightness we see of the Milky way is more evened out. A lot of the light in that white streak we see is coming from relatively close stars. It's a very different perspective than our view of Andromeda where the center stands as much brighter.
However, as I understood it, the disc (as well as the center) are the areas of highest concentration of stars.
You can see it gets a bit thicker in the center and the brightest part is the lower-center. The bright center is there in the picture, it's just largely blocked by gas and dust.
Are these 'gaps' caused by dark matter?
Dark matter is essentially invisible to all light, unless there's enough of it to bend light, then it can be observed indirectly.
The most basic reason is that the human naked eye cannot see all parts of the electromagnetic spectrum. There are a lot more stars than what we see on a regular basis, probably lying in the infrared region.
Not the answer you're looking for? Browse other questions tagged milky-way photography or ask your own question.
How are the photos of the Milky Way taken? | CommonCrawl |
Volume 23, Number 2 (1995), 852-878.
We consider the concentration of measure for $n$ i.i.d., two-dimensional random variables under the conditioning that they form a record. Under mild conditions, we show that all random variables tend to concentrate, as $n \rightarrow \infty$, around limiting curves, which are the solutions of an appropriate variational problem. We also show that the same phenomenon occurs, without the records conditioning, for the longest increasing subsequence in the sample.
Ann. Probab., Volume 23, Number 2 (1995), 852-878. | CommonCrawl |
What does it take to design a 4th order band-pass filter with -24dB/Octave?
Is it accurate to say in digital audio that, when a fader is down, then its value is "-$\infty$"?
Detecting a whistle in audio clip or stream?
Given a log-plot of frequency-magnitude-phase how to apply it as an EQ curve to a signal?
I found these files on Internet: 1.wav 2.wav 3.wav 4.wav How to generate such sounds? Geno Chen wrote "Acoustic Grand Piano (Instrument #0)" What to do next?
How is temporal pre masking possible?
How to calculate arbitrary phase shift in discrete signal?
Can the Tonality index in psychoacoustics model be positive?
Is it possible to "equalise" a signal by deconvolving the impulse response of the room in which it is to be played? | CommonCrawl |
Zhihui Yang, Fan Lin, Claudia S Robertson and Kevin K W Wang.
Dual vulnerability of TDP-43 to calpain and caspase-3 proteolysis after neurotoxic conditions and traumatic brain injury.. Journal of cerebral blood flow and metabolism : official journal of the International Society of Cerebral Blood Flow and Metabolism, 2014.
Abstract Transactivation response DNA-binding protein 43 (TDP-43) proteinopathy has recently been reported in chronic traumatic encephalopathy, a neurodegenerative condition linked to prior history of traumatic brain injury (TBI). While TDP-43 appears to be vulnerable to proteolytic modifications under neurodegenerative conditions, the mechanism underlying the contribution of TDP-43 to the pathogenesis of TBI remains unknown. In this study, we first mapped out the calpain or caspase-3 TDP-43 fragmentation patterns by in vitro protease digestion. Concurrently, in cultured cerebrocortical neurons subjected to cell death challenges, we identified distinct TDP-43 breakdown products (BDPs) of 35, 33, and 12 kDa that were indicative of dual calpain/caspase attack. Cerebrocortical culture incubated with calpain and caspase-fragmented TDP-43 resulted in neuronal injury. Furthermore, increased TDP-43 BDPs as well as redistributed TDP-43 from the nucleus to the cytoplasm were observed in the mouse cortex in two TBI models: controlled cortical impact injury and overpressure blast-wave-induced brain injury. Finally, TDP-43 and its 35 kDa fragment levels were also elevated in the cerebrospinal fluid (CSF) of severe TBI patients. This is the first evidence that TDP-43 might be involved in acute neuroinjury and TBI pathology, and that TDP-43 and its fragments may have biomarker utilities in TBI patients.Journal of Cerebral Blood Flow & Metabolism advance online publication, 11 June 2014; doi:10.1038/jcbfm.2014.105.
Mie Kubota-Sakashita, Kazuya Iwamoto, Miki Bundo and Tadafumi Kato.
A role of ADAR2 and RNA editing of glutamate receptors in mood disorders and schizophrenia.. Molecular brain 7:5, January 2014.
Abstract BACKGROUND: Pre-mRNAs of 2-amino-3-(3-hydroxy-5-methyl-isoxazol-4-yl)-propanoic acid (AMPA)/kainate glutamate receptors undergo post-transcriptional modification known as RNA editing that is mediated by adenosine deaminase acting on RNA type 2 (ADAR2). This modification alters the amino acid sequence and function of the receptor. Glutamatergic signaling has been suggested to have a role in mood disorders and schizophrenia, but it is unknown whether altered RNA editing of AMPA/kainate receptors has pathophysiological significance in these mental disorders. In this study, we found that ADAR2 expression tended to be decreased in the postmortem brains of patients with schizophrenia and bipolar disorder. RESULTS: Decreased ADAR2 expression was significantly correlated with decreased editing of the R/G sites of AMPA receptors. In heterozygous Adar2 knockout mice (Adar2+/- mice), editing of the R/G sites of AMPA receptors was decreased. Adar2+/- mice showed a tendency of increased activity in the open-field test and a tendency of resistance to immobility in the forced swimming test. They also showed enhanced amphetamine-induced hyperactivity. There was no significant difference in amphetamine-induced hyperactivity between Adar2+/- and wild type mice after the treatment with an AMPA/kainate receptor antagonist, 2,3-dihydroxy-6-nitro-7-sulfamoyl-benzo[f]quinoxaline. CONCLUSIONS: These findings collectively suggest that altered RNA editing efficiency of AMPA receptors due to down-regulation of ADAR2 has a possible role in the pathophysiology of mental disorders.
Takenari Yamashita and Shin Kwak.
The molecular link between inefficient GluA2 Q/R site-RNA editing and TDP-43 pathology in motor neurons of sporadic amyotrophic lateral sclerosis patients.. Brain research, 2013.
Abstract TAR DNA-binding protein (TDP-43) pathology and reduced expression of adenosine deaminase acting on RNA 2 (ADAR2), which is the RNA editing enzyme responsible for adenosine-to-inosine conversion at the GluA2 glutamine/arginine (Q/R) site, concomitantly occur in the same motor neurons of amyotrophic lateral sclerosis (ALS) patients; this finding suggests a link between these two ALS-specific molecular abnormalities. AMPA receptors containing Q/R site-unedited GluA2 in their subunit assembly are Ca(2+)-permeable, and motor neurons lacking ADAR2 undergo slow death in conditional ADAR2 knockout (AR2) mice, which is a mechanistic ALS model in which the ADAR2 gene is targeted in cholinergic neurons. Moreover, deficient ADAR2 induced mislocalization of TDP-43 similar to TDP-43 pathology seen in the sporadic ALS patients in the motor neurons of AR2 mice. The abnormal mislocalization of TDP-43 specifically resulted from activation of the Ca(2+)-dependent serine protease calpain that specifically cleaved TDP-43 at the C-terminal region, and generated aggregation-prone N-terminal fragments. Notably, the N-terminal fragments of TDP-43 lacking the C-terminus were demonstrated in the brains and spinal cords of ALS patients. Because normalization of either the Ca(2+)-permeability of AMPA receptors or the calpain activity in the motor neurons normalized the subcellular localization of TDP-43 in AR2 mice, it is likely that exaggerated calpain-dependent TDP-43 fragments played a role at least in the initiation of TDP-43 pathology. Elucidation of the molecular cascade of neuronal death induced by ADAR2 downregulation could provide a new specific therapy for sporadic ALS. In this review, we summarized the work from our group on the role of inefficient GluA2 Q/R site-RNA editing and TDP-43 pathology in sporadic ALS, and discussed possible effects of inefficient ADAR2-mediated RNA editing in general. This article is part of a Special Issue entitled RNA Metabolism 2013.
Takenari Yamashita, Hui Lin Chai, Sayaka Teramoto, Shoji Tsuji, Kuniko Shimazaki, Shin-ichi Muramatsu and Shin Kwak.
Rescue of amyotrophic lateral sclerosis phenotype in a mouse model by intravenous AAV9-ADAR2 delivery to motor neurons.. EMBO molecular medicine 5(11):1710–9, 2013.
Abstract Amyotrophic lateral sclerosis (ALS) is the most common adult-onset motor neuron disease, and the lack of effective therapy results in inevitable death within a few years of onset. Failure of GluA2 RNA editing resulting from downregulation of the RNA-editing enzyme adenosine deaminase acting on RNA 2 (ADAR2) occurs in the majority of ALS cases and causes the death of motor neurons via a Ca(2+) -permeable AMPA receptor-mediated mechanism. Here, we explored the possibility of gene therapy for ALS by upregulating ADAR2 in mouse motor neurons using an adeno-associated virus serotype 9 (AAV9) vector that provides gene delivery to a wide array of central neurons after peripheral administration. A single intravenous injection of AAV9-ADAR2 in conditional ADAR2 knockout mice (AR2), which comprise a mechanistic mouse model of sporadic ALS, caused expression of exogenous ADAR2 in the central neurons and effectively prevented progressive motor dysfunction. Notably, AAV9-ADAR2 rescued the motor neurons of AR2 mice from death by normalizing TDP-43 expression. This AAV9-mediated ADAR2 gene delivery may therefore enable the development of a gene therapy for ALS.
Christopher S Brower, Konstantin I Piatkov and Alexander Varshavsky.
Neurodegeneration-associated protein fragments as short-lived substrates of the N-end rule pathway.. Molecular cell 50(2):161–71, 2013.
Abstract Protein aggregates are a common feature of neurodegenerative syndromes. Specific protein fragments were found to be aggregated in disorders including Alzheimer's disease, amyotrophic lateral sclerosis, and Parkinson's disease. Here, we show that the natural C-terminal fragments of Tau, TDP43, and $\alpha$-synuclein are short-lived substrates of the Arg/N-end rule pathway, a processive proteolytic system that targets proteins bearing "destabilizing" N-terminal residues. Furthermore, a natural TDP43 fragment is shown to be metabolically stabilized in Ate1(-/-) fibroblasts that lack the arginylation branch of the Arg/N-end rule pathway, leading to accumulation and aggregation of this fragment. We also found that a fraction of A$\beta$42, the Alzheimer's disease-associated fragment of APP, is N-terminally arginylated in the brains of 5xFAD mice and is degraded by the Arg/N-end rule pathway. The discovery that neurodegeneration-associated natural fragments of TDP43, Tau, $\alpha$-synuclein, and APP can be selectively destroyed by the Arg/N-end rule pathway suggests that this pathway counteracts neurodegeneration.
Liu Yang, Ping Huang, Feng Li, Liyun Zhao, Yongliang Zhang, Shoufeng Li, Zhenji Gan, Anning Lin, Wenjun Li and Yong Liu.
c-Jun amino-terminal kinase-1 mediates glucose-responsive upregulation of the RNA editing enzyme ADAR2 in pancreatic beta-cells.. PloS one 7(11):e48611, January 2012.
Abstract A-to-I RNA editing catalyzed by the two main members of the adenosine deaminase acting on RNA (ADAR) family, ADAR1 and ADAR2, represents a RNA-based recoding mechanism implicated in a variety of cellular processes. Previously we have demonstrated that the expression of ADAR2 in pancreatic islet $\beta$-cells is responsive to the metabolic cues and ADAR2 deficiency affects regulated cellular exocytosis. To investigate the molecular mechanism by which ADAR2 is metabolically regulated, we found that in cultured $\beta$-cells and primary islets, the stress-activated protein kinase JNK1 mediates the upregulation of ADAR2 in response to changes of the nutritional state. In parallel with glucose induction of ADAR2 expression, JNK phosphorylation was concurrently increased in insulin-secreting INS-1 $\beta$-cells. Pharmacological inhibition of JNKs or siRNA knockdown of the expression of JNK1 prominently suppressed glucose-augmented ADAR2 expression, resulting in decreased efficiency of ADAR2 auto-editing. Consistently, the mRNA expression of Adar2 was selectively reduced in the islets from JNK1 null mice in comparison with that of wild-type littermates or JNK2 null mice, and ablation of JNK1 diminished high-fat diet-induced Adar2 expression in the islets from JNK1 null mice. Furthermore, promoter analysis of the mouse Adar2 gene identified a glucose-responsive region and revealed the transcription factor c-Jun as a driver of Adar2 transcription. Taken together, these results demonstrate that JNK1 serves as a crucial component in mediating glucose-responsive upregulation of ADAR2 expression in pancreatic $\beta$-cells. Thus, the JNK1 pathway may be functionally linked to the nutrient-sensing actions of ADAR2-mediated RNA editing in professional secretory cells.
Takenari Yamashita, Takuto Hideyama, Kosuke Hachiga, Sayaka Teramoto, Jiro Takano, Nobuhisa Iwata, Takaomi C Saido and Shin Kwak.
A role for calpain-dependent cleavage of TDP-43 in amyotrophic lateral sclerosis pathology.. Nature communications 3:1307, January 2012.
Abstract Both mislocalization of TDP-43 and downregulation of RNA-editing enzyme ADAR2 co-localize in the motor neurons of amyotrophic lateral sclerosis patients, but how they are linked is not clear. Here we demonstrate that activation of calpain, a Ca2+-dependent cysteine protease, by upregulation of Ca2+-permeable AMPA receptors generates carboxy-terminal-cleaved TDP-43 fragments and causes mislocalization of TDP-43 in the motor neurons expressing glutamine/arginine site-unedited GluA2 of conditional ADAR2 knockout (AR2) mice that mimic the amyotrophic lateral sclerosis pathology. These abnormalities are inhibited in the AR2res mice that express Ca2+-impermeable AMPA receptors in the absence of ADAR2 and in the calpastatin transgenic mice, but are exaggerated in the calpastatin knockout mice. Additional demonstration of calpain-dependent TDP43 fragments in the spinal cord and brain of amyotrophic lateral sclerosis patients, and high vulnerability of amyotrophic lateral sclerosis-linked mutant TDP43 to cleavage by calpain support the crucial role of the calpain-dependent cleavage of TDP43 in the amyotrophic lateral sclerosis pathology.
Takuto Hideyama, Takenari Yamashita, Hitoshi Aizawa, Shoji Tsuji, Akiyoshi Kakita, Hitoshi Takahashi and Shin Kwak.
Profound downregulation of the RNA editing enzyme ADAR2 in ALS spinal motor neurons.. Neurobiology of disease 45(3):1121–8, 2012.
Abstract Amyotrophic lateral sclerosis (ALS) is the most common adult-onset fatal motor neuron disease. In spinal motor neurons of patients with sporadic ALS, normal RNA editing of GluA2, a subunit of the L-$\alpha$-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) receptor, is inefficient. Adenosine deaminase acting on RNA 2 (ADAR2) specifically mediates RNA editing at the glutamine/arginine (Q/R) site of GluA2 and motor neurons expressing Q/R site-unedited GluA2 undergo slow death in conditional ADAR2 knockout mice. Therefore, investigation into whether inefficient ADAR2-mediated GluA2 Q/R site-editing occurs universally in motor neurons of patients with ALS would provide insight into the pathogenesis of ALS. We analyzed the extents of GluA2 Q/R site-editing in an individual laser-captured motor neuron of 29 ALS patients compared with those of normal and disease control subjects. In addition, we analyzed the enzymatic activity of three members of the ADAR family (ADAR1, ADAR2 and ADAR3) in ALS motor neurons expressing unedited GluA2 mRNA and those expressing only edited GluA2 mRNA. Q/R site-unedited GluA2 mRNA was expressed in a significant proportion of motor neurons from all of the ALS cases examined. Conversely, motor neurons of the normal and disease control subjects expressed only edited GluA2 mRNA. ADAR2, but not ADAR1 or ADAR3, was significantly downregulated in all the motor neurons of ALS patients, more extensively in those expressing Q/R site-unedited GluA2 mRNA than those expressing only Q/R site-edited GluA2 mRNA. These results indicate that ADAR2 downregulation is a profound pathological change relevant to death of motor neurons in ALS.
Commentary (two hits with one shot: a possibility of simultaneous targeting motor neuron loss and depression in ALS by upregulating ADAR2).. CNS & neurological disorders drug targets 10(8):863, December 2011.
Baoman Li, Shiquen Zhang, Hongyan Zhang, Leif Hertz and Liang Peng.
Fluoxetine affects GluK2 editing, glutamate-evoked Ca(2+) influx and extracellular signal-regulated kinase phosphorylation in mouse astrocytes.. Journal of psychiatry & neuroscience : JPN 36(5):322–38, 2011.
Abstract BACKGROUND: We sought to study the effects of chronic exposure to fluoxetine - a selective serotonin reuptake inhibitor (SSRI) and specific 5-HT(2B) receptor agonist in astrocytes - on the expression of kainate receptors (GluK1-5) in cultured astrocytes and in intact brains in mice and on GluK2 editing by adenosine deaminase acting on RNA (ADAR), as well as the ensuing effects of fluoxetine on glutamate-mediated Ca(2+) influx and extracellular signal-regulated kinase (ERK)(1/2) phosphorylation in astrocytes. METHODS: We performed reverse transcription-polymerase chain reaction (PCR) to assess mRNA expression. We analyzed RNA editing with amplification refractory mutation system PCR and complementary DNA sequencing. Protein expression and ERK phosphorylation were assessed using Western blots. We studied gene silencing with specific small interfering RNAs (siRNA), and we studied intracellular Ca(2+) using fluorometry. RESULTS: All GluK subunits were present in the brain in vivo, and GluK2-5 subunits were present in cultured astrocytes. Fluoxetine upregulated GluK2 and ADAR2. Enhanced GluK2 editing by fluoxetine abolished glutamate-mediated increases in intra cellular Ca(2+) and ERK(1/2) phosphorylation. Enhanced editing of GluK2 was prevented by siRNA against the 5-HT(2B) receptor or ADAR2. LIMITATIONS: Limitations of our study include the use of an in vitro system, but our cultured cells in many respects behave like in vivo astrocytes. CONCLUSION: Fluoxetine alters astrocytic glutamatergic function.
Takuto Hideyama and Shin Kwak.
When Does ALS Start? ADAR2-GluA2 Hypothesis for the Etiology of Sporadic ALS.. Frontiers in molecular neuroscience 4:33, 2011.
Abstract Amyotrophic lateral sclerosis (ALS) is the most common adult-onset motor neuron disease. More than 90% of ALS cases are sporadic, and the majority of sporadic ALS patients do not carry mutations in genes causative of familial ALS; therefore, investigation specifically targeting sporadic ALS is needed to discover the pathogenesis. The motor neurons of sporadic ALS patients express unedited GluA2 mRNA at the Q/R site in a disease-specific and motor neuron-selective manner. GluA2 is a subunit of the AMPA receptor, and it has a regulatory role in the Ca(2+)-permeability of the AMPA receptor after the genomic Q codon is replaced with the R codon in mRNA by adenosine-inosine conversion, which is mediated by adenosine deaminase acting on RNA 2 (ADAR2). Therefore, ADAR2 activity may not be sufficient to edit all GluA2 mRNA expressed in the motor neurons of ALS patients. To investigate whether deficient ADAR2 activity plays pathogenic roles in sporadic ALS, we generated genetically modified mice (AR2) in which the ADAR2 gene was conditionally knocked out in the motor neurons. AR2 mice showed an ALS-like phenotype with the death of ADAR2-lacking motor neurons. Notably, the motor neurons deficient in ADAR2 survived when they expressed only edited GluA2 in AR2/GluR-B(R/R) (AR2res) mice, in which the endogenous GluA2 alleles were replaced by the GluR-B(R) allele that encoded edited GluA2. In heterozygous AR2 mice with only one ADAR2 allele, approximately 20% of the spinal motor neurons expressed unedited GluA2 and underwent degeneration, indicating that half-normal ADAR2 activity is not sufficient to edit all GluA2 expressed in motor neurons. It is likely therefore that the expression of unedited GluA2 causes the death of motor neurons in sporadic ALS. We hypothesize that a progressive downregulation of ADAR2 activity plays a critical role in the pathogenesis of sporadic ALS and that the pathological process commences when motor neurons express unedited GluA2.
Shuchen Lee, Guang Yang, Yue Yong, Ying Liu, Liyun Zhao, Jing Xu, Xiaomin Zhang, Yanjie Wan, Chun Feng, Zhiqin Fan, Yong Liu, Jia Luo and Zun-Ji Ke.
ADAR2-dependent RNA editing of GluR2 is involved in thiamine deficiency-induced alteration of calcium dynamics.. Molecular neurodegeneration 5:54, January 2010.
Abstract BACKGROUND: Thiamine (vitamin B1) deficiency (TD) causes mild impairment of oxidative metabolism and region-selective neuronal loss in the central nervous system (CNS). TD in animals has been used to model aging-associated neurodegeneration in the brain. The mechanisms of TD-induced neuron death are complex, and it is likely multiple mechanisms interplay and contribute to the action of TD. In this study, we demonstrated that TD significantly increased intracellular calcium concentrations [Ca2+]i in cultured cortical neurons. RESULTS: TD drastically potentiated AMPA-triggered calcium influx and inhibited pre-mRNA editing of GluR2, a Ca2+-permeable subtype of AMPA receptors. The Ca2+ permeability of GluR2 is regulated by RNA editing at the Q/R site. Edited GluR2 (R) subunits form Ca2+-impermeable channels, whereas unedited GluR2 (Q) channels are permeable to Ca2+ flow. TD inhibited Q/R editing of GluR2 and increased the ratio of unedited GluR2. The Q/R editing of GluR2 is mediated by adenosine deaminase acting on RNA 2 (ADAR2). TD selectively decreased ADAR2 expression and its self-editing ability without affecting ADAR1 in cultured neurons and in the brain tissue. Over-expression of ADAR2 reduced AMPA-mediated rise of [Ca2+]i and protected cortical neurons against TD-induced cytotoxicity, whereas down-regulation of ADAR2 increased AMPA-elicited Ca2+ influx and exacerbated TD-induced death of cortical neurons. CONCLUSIONS: Our findings suggest that TD-induced neuronal damage may be mediated by the modulation of ADAR2-dependent RNA Editing of GluR2.
Jun Sawada, Takenari Yamashita, Hitoshi Aizawa, Yoko Aburakawa, Naoyuki Hasebe and Shin Kwak.
Effects of antidepressants on GluR2 Q/R site-RNA editing in modified HeLa cell line.. Neuroscience research 64(3):251–8, 2009.
Abstract Marked reduction of RNA editing at the glutamine (Q)/arginine (R) site of the glutamate receptor subunit type 2 (GluR2) in motor neurons may be a contributory cause of neuronal death specifically in sporadic ALS. It has been shown that deregulation of RNA editing of several mRNAs plays a causative role in diseases of the central nervous system such as depression. We analyzed the effects of eight antidepressants on GluR2 Q/R site-RNA editing in a modified HeLa cell line that stably expresses half-edited GluR2 pre-mRNA. We also measured changes in RNA expression levels of adenosine deaminase acting on RNA type 2 (ADAR2), the specific RNA editing enzyme of the GluR2 Q/R site, and GluR2, in order to assess the molecular mechanism causing alteration of this site-editing. The editing efficiency at the GluR2 Q/R site was significantly increased after treatment with seven out of eight antidepressants at a concentration of no more than 10 microM for 24h. The relative abundance of ADAR2 mRNA to GluR2 pre-mRNA or to beta-actin mRNA was increased after treatment with six of the effective antidepressants, whereas it was unchanged after treatment with milnacipran. Our results suggest that antidepressants have the potency to enhance GluR2 Q/R site-editing by either upregulating the ADAR2 mRNA expression level or other unidentified mechanisms. It may be worth investigating the in vivo efficacy of antidepressants with a specific therapeutic strategy for sporadic ALS in view. | CommonCrawl |
That's all equivalences I can see so far. But they all seem so similar. Are they all equivalent? If not, is there a subset of the set of the above theorems the elements of which are pairwise equivalent?
Browse other questions tagged real-analysis linear-algebra differential-geometry differential-topology multilinear-algebra or ask your own question.
How to find the basis of the Col($A$) from the basis of Null($A$)?
Why is every element of this group upper triangular?
How would you define the volume of a parallelepiped in $\mathbb R^k$? | CommonCrawl |
Volume 7, Number 10 (2002), 1193-1214.
We consider a semilinear problem of the type $Lu=f(b,u),$ where $f(b,u)\simeq bu$ as $u\to 0$ and $f(b,u)\simeq b_\infty u$ as $\|u \| \to\infty$ assuming that there exist a finite number of eigenvalues of the linear operator $L$ between $b$ and $b_\infty$. Under suitable assumptions we prove the existence of four nontrivial solutions for $b$ close to an eigenvalue. We give an application to problems of oscillations of a forced beam.
Adv. Differential Equations, Volume 7, Number 10 (2002), 1193-1214. | CommonCrawl |
The spin-Hall effect (SHE) and its reciprocal, the inverse spin-Hall effect (ISHE), are of great importance in spintronics since they enable, respectively, the conversion of a longitudinal charge current to a transverse spin current and the reverse process. Here we will report on a ferromagnetic resonance (FMR) study of FeCoB/W thin film bi-layer structures that incorporate different W thicknesses and hence difference phases. A very large negative spin Hall angle has been observed in the $\beta $-W samples and confirmed by spin-torque switching studies. Alternatively FMR measurements with bilayers containing $\alpha $-W suggests a strong positive SHE, but this interpretation of the experiment is not consistent with spin torque switching studies utilizing $\alpha $-W. Since the $\alpha $-W FMR results also show an enhanced magnetic damping we tentatively attribute these results to a significantly enhanced spin pumping effect in $\alpha $-W, relative to $\beta $-W. Magnetization measurements indicate that the two different types of FeCoB/W bilayers have substantially different interfacial magnetic anisotropy coefficients. We will discuss these results, together with the differing temperature dependence of the FMR signal in the two cases, which help point the way to understanding the origin of the giant SHE in $\beta $-W and the strong ISHE in $\alpha $-W. | CommonCrawl |
Theorem 1: Let $A$ and $B$ be countable sets. Then the cartesian product $A \times B$ is countable.
Proof: There are three cases to consider.
Case 1: If both $A$ and $B$ are finite with $|A| = m$ and $|B| = n$ then it is easy to show that $|A \times B| = mn$ and hence $A \times B$ is finite and so it is countable.
Then clearly $h$ is injective by the unique factorization of the each natural number. So $A \times B$ is countable. | CommonCrawl |
Move $1$ match and make this correct.
You can't break the match.
You can't make an inequality sign, just change numbers and/or operators.
moving the cross stick in the plus sign to go diagonally across the equals sign.
And there were no matches (hehe).
just change the equals sign to a more than sign!
If only the first 5 would be a 6. That would help a lot. Please ask your sister for the solution and double check if it's actually possible. Because I have a hunch that this puzzle is impossible.
I saw this in "hot network questions" and tried to solve it on paper before I clicked to avoid getting spoiled. So I scribbled down the equation. When I couldn't find a way to solve it, I finally opened the question and saw, that my 7 had one less matchstick (the very left one). So I wondered about the display of numbers we don't get to see. Would a 9 without an underscore be legal for example? It clearly would be a distinct nine, wheter the bottom stick is there or not. You wouldn't return your 80's alarm clock because of such a nine, anyway.
37 + 32 = 69 by taking the bottom matchstick from the second five to make the first five a six. Granted, the resulting nine is somewhat weird, but you clearly would not assume another number instead of it. Maybe for arguments sake just now, but not if some hot girl wrote the nines of her phone number in that way. That would probably just be a-okay for you. So just give me the correct flag now. Thanks.
Take the vertical stick in the $+$ turning it into a $-$ and put left-below the first $5$ to get $_15\,5$. Interpret this as $1^55=5$.
Remove the match on the + to make it a -, then eat it. Then crop the picture so that it cuts out the last 5. 37 - 32 = 5, only one match (and the frame of the picture) has moved.
Take a match from the equal sign and put it anywhere else where it creates a number. For example: 37 + 92 - 55. It's neither true nor false and it's left to the reader to calculate!
How about a hexadecimal answer?
Take one of the $=$ matches, break it in half, and put it on the right side of the remaining = match to make a right facing arrow $\rightarrow$. Then, the LHS is $69$, interpreted as a boolean is True, and the RHS is $55$, interpreted as a boolean is True. Then, the statment True $\rightarrow$ True is True. Thus, the statement becomes True.
Make use of the fact that those are not arbitrary sticks, but matches. So, take the vertical match from the plus (making it a minus), move it quickly over the side of the match box (so it starts burning), and then move it in turn to all the matches making up the left digits (so they all catch fire and burn away, without moving). Then put that match anywhere out of the way to finish its burning without affecting the rest of the matches.
The remaining matches form the equation 7-2=5.
37 + 3P = 55 where P = 6.
But all of those require more than one match. I'll keep at it, if I solve it I'll update. All in all, I have found 210 total numeric combinations. None of which are achieved with a single move. I have written a loop in C# that goes through multiple arrays of all possible number combinations to confirm this. I may be missing something, but mathematically; this seems quite impossible, aside from the Cheater-Pants, you can't do that solutions that El-Guest and several others (including myself) have posed.
37 + 3P = 55 evaluates true when P = 6; this breaks down to 37 + 18 = 55.
If you down vote, please explain why the down vote is justified.
The inequality is technically correct, which is the best kind of correct.
You can either interpret that as a zero or a five that's been slashed out.
My answer, similar to @alto's, is a little bit more elegant.
Take the leftmost downstroke from the 7 and place it at an angle above and and touching the top bar of the = sign to form 'a greater-than-or-equal' (or less-than-or-equal") sign.
leftmost downstroke); 6 and 9 (by adding a horizontal top/bottom stroke).
OneCharacterMoves = # match changed position inside one character, e.g. change "3' to '2', "6" to "9" etc.
Inspecting the values shows that none of them can be produced by moving one matchstick.
Here is a solution exploiting that the algebraic operations are not specified in the question.
Take the vertical part of the plus sign and place it horizontally above the equality sign to obtain 37 "minus" 32 "is defined as" 55.
Just remove the vertical match from the "+" sign and add onto 32, now it is 92.
Move the lower match of the plus and the top match moves too! Make $37\times32=1184=550\times2+84$, and the picture has been (unfairly in my opinion) cropped so that you can't see the last bit!
Remove the lower right matchstick to give 3 to the power of pi. Break the stick a bit and the extra matchstick on the equals to give an approximately equals symbol.
37 - 32 = 5 a clue is that in second 5 match heads are aligned so that they burn each other properly until 5 is fully burnt, so perhaps we are supposed to ignite it with the match we took away.
Remove the vertical match from the +, break it in half, put one half on the upper left of the first 3 to make it a 9, and the other half in front of the first 5 as a minus sign.
I know the way I constructed this pattern doesn't match the original picture, but I'm sure I nailed the right pattern.
$$3^1 \times 7 + 32 = 53$$ ...just turn the $55$ into a $53$ by moving one match.
Edit: Sorry, I didn't notice that Kamil posted the same answer earlier.
moved the vertical bar of the $+$ sign moved to the $=$ sign to get a triple bar symbol. This turns the expression into a modular arithmetic expression.
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How many matches can you fit? | CommonCrawl |
Abstract: We prove that the metric projection onto a finite-dimensional subspace $Y\subset L_p$, $p\in(1,2)\cup(2,\infty)$, satisfies the Lipschitz condition if and only if every function in $Y$ is supported on finitely many atoms. We estimate the Lipschitz constant of such a projection for the case in which the subspace is one-dimensional.
Keywords: metric projection, Lipschitz condition, $L_p$ space, linearity coefficient.
The work of all the authors was supported by the Russian Foundation for Basic Research under grant no. 15-01-08335. The first author's work was supported by Dmitry Zimin's "Dynasty" foundation. | CommonCrawl |
Statistical analysis of familial correlations.
Inferences for familial (intraclass and interclass) correlations are considered for unbalanced familial data from multivariate normal populations. The most commonly used familial correlation is that between siblings (without regard to gender), called intraclass or sib-sib correlation coefficient. The expressions for the large sample biases and variances of several point estimators of the intraclass correlation are derived and the sampling properties of these estimators are compared for a wide variety of unbalanced designs. It is recommended that the Karlin's individual estimator should be used for the small number of groups with a severe degree of unbalancedness. However, for a large number of groups, the Karlin's empirical estimator is recommended provided that the true value of intraclass correlation is less than or equal to 0.5. Several procedures for testing that the intraclass correlation is equal to a specified value are derived and compared by extensive Monte Carlo studies. The Neyman's C($\alpha$) (or partial score) and modified F-ANOVA procedures are shown to be consistently more powerful than the other procedures for the said hypotheses. Estimation procedures, based on the maximum likelihood and the ANOVA methods, for intraclass correlations in multiple samples are discussed. In order to test the homogeneity of the intraclass correlations in multiple samples, several procedures are derived and compared in terms of their empirical powers. The use of a test based on Fisher's variance stabilizing transformation is recommended for small values of the common intraclass correlation and of the Neyman's C($\alpha$) test for moderate values of the common intraclass correlation is recommended. The maximum likelihood estimation of sibling correlations (brother-brother, sister-sister, and brother-sister correlations) is considered next and it is shown that the estimates of the parameters can be obtained by numerical maximization of a function of fewer parameters. The expressions for the large sample variances and covariances of the estimators are derived and the procedures to test the significance of these correlations are discussed. The procedures are illustrated by using a published arterial blood pressures dataset from the literature. Using a linear model approach, procedures to find the maximum likelihood estimates of five familial correlations (mother-brother, mother-sister, brother-brother, sister-sister and brother-sister correlations) and other parameters are developed. The expressions for the asymptotic variances and covariances of estimators are derived. Procedures for testing the significance of the above familial correlations are also presented and the methodologies are illustrated on the previously mentioned epidemiological data sets.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1989 .M535. Source: Dissertation Abstracts International, Volume: 52-11, Section: B, page: 5915. Co-Supervisors: Mohamed M. Shoukri; Derrick S. Tracy. Thesis (Ph.D.)--University of Windsor (Canada), 1989.
Mian, Ijaz Ul Hassan., "Statistical analysis of familial correlations." (1989). Electronic Theses and Dissertations. 3580. | CommonCrawl |
Is it appropriate to use a Bonferroni adjustment in all cases of multiple testing?
If one performs a test on a data set, then one splits that data set into finer levels (e.g. split the data by gender) and performs the same tests, how might this affect the number of individual tests that are perceived? That is, if X hypotheses are tested on a dataset containing data from both males and females and then the dataset is split to give male and female data separately and the same hypotheses tested, would the number of individual hypotheses remain as X or increase due to the additional testing?
The Bonferroni adjustment will always provide strong control of the family-wise error rate. This means that, whatever the nature and number of the tests, or the relationships between them, if their assumptions are met, it will ensure that the probability of having even one erroneous significant result among all tests is at most $\alpha$, your original error level. It is therefore always available.
Whether it is appropriate to use it (as opposed to another method or perhaps no adjustment at all) depends on your objectives, the standards of your discipline and the availability of better methods for your specific situation. At the very least, you should probably consider the Holm-Bonferroni method, which is just as general but less conservative.
Regarding your example, since you are performing several tests, you are increasing the family-wise error rate (the probability of rejecting at least one null hypothesis erroneously). If you only perform one test on each half, many adjustments would be possible including Hommel's method or methods controlling the false discovery rate (which is different from the family-wise error rate). If you conduct a test on the whole data set followed by several sub-tests, the tests are no longer independent so some methods are no longer appropriate. As I said before, Bonferroni is in any case always available and guaranteed to work as advertised (but also to be very conservative…).
You could also just ignore the whole issue. Formally, the family-wise error rate is higher but with only two tests it's still not so bad. You could also start with a test on the whole data set, treated as the main outcome, followed by sub-tests for different groups, uncorrected because they are understood as secondary outcomes or ancillary hypotheses.
If you consider many demographic variables in that way (as opposed to just planning to test for gender differences from the get go or perhaps a more systematic modeling approach), the problem becomes more serious with a significant risk of "data dredging" (one difference comes out significant by chance allowing you to rescue an inconclusive experiment with some nice story about the demographic variable to boot whereas in fact nothing really happened) and you should definitely consider some form of adjustment for multiple testing. The logic remains the same with X different hypotheses (testing X hypotheses twice – one on each half of the data set – entails a higher family-wise error rate than testing X hypotheses only once and you should probably adjust for that).
To be fair, I have looked at many different economic/ econometric articles for my current research project and in that limited experience I haven't come across many articles applying such corrections when comparing 2-5 tests.
You must remember that medical data and scientific data are irreconcilably different in that heteroscedastic medical data is never experimental unlike homoscedastic biological data. Recall also that many discussions on role of power testing and Bonferroni type corrections involve only speculations on the nature of unknowable alternate distribtions. Setting beta in a power calculation is an arbitrary procedure. None of the medical statisticians advertise this. Second, if there is autocorrelation of (within) data samples the Central Limit Theorem has been violated and Normal based Gaussian testing is not valid. Third, recall that the Normal Distribution is becoming outmoded in the sense that many medical phenomena are fractal based distributions that possess neither finite means and/or finite variances (Cauchy-type distributions) and require fractal resistant statistical analyses. Carrying out any post-hoc anslysis drpending on what you find during early analysis is improper. Finally, between-subject bijectivity is not necessarily valid and the conditions for Bonferroni corrections are important elements to be uniquely teased out during a priori Experimental Design only. Nigel T. James. MB BChir, (UK medical degrees), MSc (in Applied Statistics).
Not the answer you're looking for? Browse other questions tagged multiple-comparisons bonferroni type-i-and-ii-errors or ask your own question.
Why do I need a Bonferroni correction if we are evaluating our primary outcome?
What's wrong with Bonferroni adjustments?
Bonferroni adjustment according to which families?
Is this an example of a multiple comparisons problem? | CommonCrawl |
This is the third post on my Hexapod Project series. The point of this project is to build a robot that allows me to try out a few robotics concepts. For a listing of each post in this series, click here. In this post, I'll talk about using laser ranging to let the hexapod keep track of where it is relative to its environment.
It's not hard to find examples of robots failing to perform tasks that a human might deem trivial. Even the entrants to the DARPA Robotic Challenge--while amazingly sophisticated in their design--sometimes fail to turn handles, walk through doors, or even just stand upright for very long. If you are unfamiliar with the subtle challenges of robotics, these failures might make you wonder why it takes so much money and time to create such a flawed system. If a computer can calculate the N-millionth digit of pi in a few milliseconds, why can't it handle walking in a straight line? The general answer to this is that the fundamental differences between the human mind and a computer cpu (neurons vs transistors, programming vs evolution, etc) create vast differences in the intrinsic difficulty of many tasks.
One such task is spatial awareness and localization. Humans are capable of integrating various senses (sight, touch, balance) into a concept of movement and location relative to an environment. To make my hexapod robot capable of autonomous navigation, it also needs to have a sense of location so that it can decide where it needs to go and where it should avoid going.
Arguably the most common way of letting a robot know where it is in the world is GPS. Satellites in orbit around the earth broadcast their position and the current time, and a GPS receiver receives these broadcasts and triangulates its position. A GPS-enabled robot can figure out its location on Earth to within a few feet or better (depending on how much money you spend on a receiver). The biggest issue with GPS is that a robot using GPS needs a clear view of the sky so that it can receive the signals being beamed down from the GPS satellites. GPS also doesn't give you any information about your surroundings, so it's impossible to navigate around obstacles.
For the hexapod, I wanted to avoid using GPS altogether. I chose to use a LIDAR unit for indoor localization, mostly because it seemed like an interesting challenge. LIDAR uses visible light pulses to measure the distance to objects just like RADAR uses radio waves bouncing off objects. A LIDAR unit contains a laser emitter/detector pair that can be swept across a scene to make measurements at different angles. At each angle, the unit emits a pulse of laser light and looks for a reflected pulse with the detector. The delay between emission and detection (and the speed of light) gives the distance to the reflecting object at that angle. High-quality LIDAR units (like those used in self-driving vehicles) can quickly give an accurate 3D scan of the surrounding environment.
The LIDAR unit I picked is taken from the XV-11 robotic vacuum cleaner from Neato Robotics. You can find just the LIDAR unit by itself as a replacement item on various sites; I got mine off eBay for around $80. The XV-11 LIDAR unit has a bit of a following in the hacking community, as it offers 360-degree laser ranging at 5 Hz for much cheaper than anyone else. While the unit isn't open source, there are some nice resources online that provide enough documentation to get started. It only scans in a 2D plane, but what you lose in dimensionality you gain in monetary savings.
Laser module in the lower left, sensor and optics on the upper right.
It's hard not to trust any device that sends a smiley face as an opening line. The introduction message isn't too informative, but the fact that it is sent as plain ASCII is comforting. The unit doesn't send any laser ranging data until it is spinning at close to 5 Hz, and the processor has no way of controlling the spin motor. By applying an average of around 3 V to the motor (I did PWM from a 12 V line), the unit spins up and raw data starts flooding down the line. The resources link above provides documentation for how the data packets are formatted, but the key points are that they contain some number of laser ranging measurements, error codes, and the current spin rate of the unit. This measured spin rate can be fed into a feedback loop for the motor controller so that it stays spinning at a nice constant 5 Hz.
I decided to have the Arduino Due on the hexapod handle communication with the LIDAR unit and keeping the motor spinning at the correct rate. The Due already handles communication between the main cpu and the Arbotix-M, so what's one more device? I soldered up a simple board that included an N-channel MOSFET for PWM control of the motor, and a LM317 voltage regulator to provide the LIDAR processor with around 3.3 V.
Motor connects on the left, LIDAR controller on the right. Bottom connects to Due.
The hexapod kit came with a mounting board for adding accessory hardware, but the mounting holes on the LIDAR didn't match up. I 3D-printed a little bracket to both attach the unit to the hexapod body and provide a little space for the board I had just made.
Credit to my Single Pixel Camera project for making me buy a better printer.
Attached to the top mounting panel of the hexapod.
With the small interface board mounted below the LIDAR unit, I connected everything up to the Due. A PWM line from the Due is used to control the speed of the motor, and the Serial2 port is used to receive data from the LIDAR processor. The 12 V needed to power the motor and processor come from whatever source already powers the main UDOO board. In testing, this was either an AC adapter or a 3-cell lipo battery.
I have no idea what I'm testing, but damn does it look technical.
Rounded object at (50,100) is my head.
Each point is a single laser ranging measurement, and they span the full 360-degrees around the unit. A map like this can be made five times a second, allowing for a pretty good update rate.
At this point, the hexapod has the ability to continually scan its surroundings with lasers and accurately determine its distance from obstacles in any direction. But we still haven't solved the problem of letting the hexapod know where it is. By looking at the plot above, we can clearly see that it was sitting near the corner of a rectangular room. If we moved the robot a few feet in any direction and looked at the new map, we would be able to see that the robot had moved, and by comparing the two plots in detail we could even measure how far it moved. As humans, we are able to do this by matching similarities between the before and after plots and spotting the differences. This is one of those tasks that is relatively easy for a human to do and very tricky for a computer to do.
Using only the LIDAR scans, we want the hexapod to be able to track its movement within its environment. By matching new scans to previous ones, we can both infer movement relative to the measured obstacles and integrate new information about obstacles measured from the new location. The process of doing so is called Simultaneous Localization and Mapping (SLAM). There are many ways of solving this problem using measurements like the LIDAR scans I have access to. Some methods involve big point clouds, some involve grids. Some are 3D, some are 2D. One of the most common traits of any SLAM algorithm that I've found is that it is complicated enough to scare away amateur robotics enthusiasts. So in keeping to my goal of writing most (if not all) of the software for my hexapod, I set out to write my own algorithm.
My algorithm is not great, but it kind of works. I decided to do a 2D grid-based SLAM algorithm because a) my LIDAR scans are only in 2D, and b) point clouds are hard to work with. As the name suggests, a SLAM algorithm involves solving two problems simultaneously: localization and mapping. My algorithm keeps a map of the surroundings in memory, and given a new LIDAR scan performs two steps: matching the new scan to the existing map and infering where the scan was measured from; and then adding the new scan to the map to update it with any changes. As the Wikipedia article on the subject suggests, we have a bit of a chicken-and-egg problem, in that you can't localize without an existing map and you can't map without knowing the location. To solve this problem, I let the hexapod know its initial coordinates and let it collect a few scans while standing still to create an initial map. Then, it is allowed to step through the full SLAM algorithm with a map already set up.
My 350 by 400 cm workroom. My head is still at (50,100).
Here, we project each LIDAR measurement ($x_i'$,$y_i'$) on to the SLAM map, adjusting for the current best guess of ($x$,$y$,$\theta$). At each projected point, we sum up the distance from that point to every occupied pixel of the SLAM map. This gives us an estimate for how 'far away' the projected scan is from matching the existing map. The three extra terms on the $\Psi$ equation are to bias the solution towards guess values for ($x$,$y$,$\theta$).
In this way, we are finding the location of the hexapod so that the new LIDAR scan looks most like the existing map. The assumption being made here is that the new scan is similar enough to the existing map that is can be matched with some confidence. Solving the above equation is a problem of non-linear optimization, similar to the inverse kinematics solved in the previous post. The code to solve this problem is a little dense, so I won't try to explain all of the details here. The relevant code is here, and the relevant method is slam::step(...);.
In words, we compute the $\Psi$ equation above and how it changes if we modify each of the parameters ($x$,$y$,$\theta$) by a small amount. Using this information, we can nudge each parameter by an amount that should get us to a lower value of $\Psi$. Since the problem is non-linear, we aren't guaranteed that this gets us to the lowest possible value, or even a lower one than before. To help make sure we end up in the right place, we initialize the solver with a guess position based on how the hexapod legs have moved recently. Since we went through so much trouble in the previous post to plan how the feet move, we might as well use that knowledge to help the localization solver. From there we iterate the nudging step over and over again with a smaller nudge until we find there is no way of nudging it to a lower value of $\Psi$. This is when we stop and say we have found the optimal values of ($x$,$y$,$\theta$). With that, the localization step is done!
For computational efficiency, I keep three versions of the SLAM map other than the normal one shown above. The first extra one is the original map convolved with a distance kernel, which at any position gives us an approximate distance to occupied pixels. The next two are the gradient of this distance map, one for the x-component and one for the y-component. These maps allow us to quickly evaluate both the $\Psi$ function and its derivatives with respect to ($x$,$y$,$\theta$). The distance map is computed in Fourier space using the convolution theorem, using threaded FFTW for computational speed. This method doesn't actually give us the correct distance measure for $\Psi$, but it's close enough for this basic algorithm.
The companion to localization is mapping. Once we have a solution to where the new scan was measured from, we need to add it to the existing SLAM map. While we have assumed the new scan is close enough to the existing map to be matched, it will have small differences due to the new measurement location that need to be incorporated so that the following scan is still similar enough to the map. In my SLAM code, the method that does the mapping step is slam::integrate(...);.
Each new laser ranging measurement from the new scan is projected on to the SLAM map given the estimated hexapod location from the localization step. The pixel below each point is set to 1.0, meaning we are fully confident that there is some object there. We then scan through every other pixel in the map and determine whether it is closer or farther away from the hexapod than the new scan measurements. If it is closer, we decrease the map value because the new scan measured something behind it, meaning it must be free of obstacles. If the pixel is farther, we leave it alone because we don't have any new information there; the new scan was blocked by something in front of it.
Once this mapping step is done, we have completed the two-part SLAM algorithm and are ready for another LIDAR scan. It's not the best or most accurate method, but it is easy to understand and can run on fairly low-level hardware in real-time. I've written the algorithm to run asynchronously from the main hexapod code, so new scans can be submitted and the hexapod can still walk around while the SLAM algorithm figures out where it is. On the UDOO's Cortex-A9, I can step a 1024x1024 map in around 2-3 seconds. With a 10 cm resolution, this gives over 100 meters of mapping. In practice, I've found that 10 cm is about the coarsest you can go in an indoor environment, but anything less than 3 cm is a waste of computing time.
I've also tested this algorithm out in real life with the hexapod. The following SLAM maps were collected by driving the hexapod around my apartment with a remote control. I started the hexapod out in one room, and it was able to walk into a different room and keep track of its position. The maps are pretty messy, but acceptable considering the simplicity of the algorithm being used.
Noisy map of my apartment.
It's not the best SLAM algorithm in the world, but it's relatively easy to understand and compute in an embedded setting. It seems to do best when the hexapod is inside a closed room and can see at least two of the walls. It has some issues keeping track of position when transitioning between rooms, mostly due to the sharp changes in the LIDAR scans when passing through a doorway. Still, it does a reasonable job at keeping track of the hexapod location within my apartment.
In the next post, I'll sort out how to get the hexapod to navigate autonomously. With the algorithms presented so far in this hexapod series, it becomes a straightforward procedure of using the SLAM maps to find optimal paths to pre-determined waypoints.
This is the second post on my Hexapod Project series. The point of this project is to build a robot that allows me to try out a few robotics concepts. For a listing of each post in this series, click here. In this post, I'll go over the steps (!) needed to get a hexapod robot walking.
At this point, I have a robot with six legs, three servos per leg, and the electronics and code to control each servo independently. But with no guidance for how to move the servos, the hexapod is useless. With their above-average leg count and sixfold symmetry, hexapods can move around in all kinds of unique ways. While dancing around is certainly a possibility for my hexapod, I'm really only interested in getting it to walk around. So to begin the process of getting it mobile, let's start with the basics of getting a robot to walk.
Here, $\theta_c$, $\theta_f$, and $\theta_t$ are the servo angles for the coxa, femur, and tibia joints, respectively, and $l_c$, $l_f$, and $l_t$ are the distances between the joints. The position and angle at which the leg is connected to the body are represented by $x_0$, $y_0$, $z_0$, and $\theta_0$. This set of equations represent the forward kinematics of the leg. Each leg has an identical set of equations, but with different values for the initial position and angle.
These equations can tell us where the foot is, given the angles of the servos, but we need to do the opposite. Unfortunately, there isn't any way to rearrange the equations above so that we can plug in the foot position and solve for the servo angles (go ahead and try!). Fortunately, this doesn't mean that it's an impossible task! The process of inverting these equations is called inverse kinematics, and I've done a project on it before. My other post explains how to go about solving an inverse kinematics problem, so if you're interested in the details, check that out.
In short, the inverse kinematic solver takes a target foot position and outputs the servo angles that it thinks are appropriate. Starting with the servo angles as they are, the algorithm uses the forward kinematic equations to see which way each servo needs to turn so that the foot ends up slightly closer to the target. It takes many small steps like this until the forward kinematics equations say the new set of servo angles put the foot in the right place. This kind of procedure has its flaws, though. Imagine you tell it to find the servo angles that put the foot a mile away? The algorithm has no way to achieve this since the legs aren't nearly that long. In situations like this, it often goes haywire, giving you a nonsensical result for the servo angles. So careful attention to the iteration procedure is important.
Impressing Alan the cat with robot weightlifting.
The six legs are broken up into two groups which trade off being the support for the body. The legs within each group lower to the ground in unison, move towards the back of the body, then lift up and move back to the front. The two groups do this exactly out of phase with each other so that there are always exactly three feet on the ground at any one point. For my hexapod, I've modified this a bit so that the three legs within each group hit the ground at slightly different times. I've done this to reduce the repetitive jolting that occurs from moving each leg simultaneously.
Notice the sharp change in direction at the start and end of when the foot is in contact with the floor. The transition to weight-bearing doesn't happen instantaneously (imagine carpeted floors), so the sudden transition when the foot goes from moving down to moving back creates problems. To create a smoother path for the feet to follow, I turned to Bezier curves.
Bézier curves are smooth functions that are completely determined by a sequence of points that I will call anchors. These anchors specify generally what shape the curve has, so tweaking the shape of the curve just involves moving around the anchor points. Going from a set of anchor points to the resulting Bezier curve involves a series of linear interpolations. Given some intended distance along the total path between 0 and 1, we start by linearly interpolating between each adjacent anchor points. So if we want to know where the Bezier curve is halfway along the path, we start by linearly interpolating halfway between each pair of adjacent anchors. If we have $N$ anchors, this gives us $N-1$ interpolated points. We then linearly interpolate again halfway between these $N-1$ points to get $N-2$ doubly-interpolated points. We continue this procedure until we are left with a single interpolated point, and this is the position of the Bezier curve at the halfway point.
The procedure for generating Bezier curves is a little difficult to describe, so I've made a little interface to help explain it. Drag the grey anchor points around to see how the Bezier curve changes, then increase the Guide Level slider to see the various levels of linear interpolation.
To make sure the hexapod stays steady when walking, I've kept the straight part of the foot path where the foot touches the ground, but set the return path to be a Bezier curve. I wrote a simple Bezier curve class to handle the computations on the hexapod.
Applying this Bezier curve stepping method to each leg in an alternating pattern gets the hexapod to walk forwards, but it can't yet handle turning. To implement turning, my first instinct was to simply adjust the amount by which each foot sweeps forward and back on each side of the body differently. This would cause one side to move forward more than the other, and the hexapod would turn. The problem with this method is that it isn't particularly physical. If you try it, you'll find that the hexapod has to drag its feet sideways to compensate for the fact that it is turning sidways but the feet only move forward and back. In order to let the hexapod turn naturally, you need to go into a different frame of reference.
If you tell all of the feet to move up towards the sky, the hexapod moves closer to the ground. Tell the feet to move down, the hexapod moves up. It can get confusing to figure out how to move the feet to get the body to move a certain way. I've found it's best to think that the hexapod body stays still in space and the ground just moves around relative to it. Then all you need to do is make sure the feet are in contact with that moving floor and they don't slide on it. For straight walking, we can just see it as a treadmill-like floor that continually moves backwards, and the feet are just trying to match the treadmill speed. For turning, we can think about the hexapod sitting above a giant turntable. How close the hexapod body sits to the axis of rotation determines how sharp of a turn it makes, or what the turning radius is. In order to keep the feet from sliding around on the turntable, we need to make sure each foot travels along a curve of constant radius from the axis of rotation. If we set it so the axis of rotation is directly underneath the body, the hexapod will stay in one place and just turn around and around. If the axis of rotation is set very very far away, there will barely be any curvature to the foot-paths, and the hexapod will basically walk forward in a straight line.
To help explain this concept, I've made another little interface for seeing how the hexapod feet need to move. Move the bottom slider around to change the turning radius, and enable the helper lines to see how each foot follows a specific path relative to the axis of rotation.
To incorporate the Bezier curve method from above into this view of walking, I convert the foot positions into polar coordinates around the axis of rotation and use the Bezier curve to pick the $\theta$ and $z$ coordinates as a function of time. In the hexapod code, I've parameterized the turning by a single parameter that relates to the turning radius. Between the turning parameter and a single speed parameter, I have full control over the movement of the hexapod.
At every time step, the code considers the current values for speed and turning, and decides where along the Bezier curve each foot should be. It then computes the actual positions in space from the curves and feeds these positions into the inverse kinematic solver. The solver outputs servo angles for each joint of each leg, and the main code packages these up and sends them off to the lower-level processors. This whole procedure is fairly quick to compute, so I can update the servo positions at about 50Hz.
At this point, the hexapod can walk around freely, but does not know where to go. In the next post, I'll go into giving the hexapod a sense of awareness through laser ranging. | CommonCrawl |
Supported by Global COE Program "The Research and Training Center for New Development in Mathematics", Graduate School of Mathematical Sciences, the University of Tokyo.
Abstract: Kazhdan's property (T) is one of the most important properties in the analytic group theory, and has a numerous applications to many other fields of pure and applied mathematics. The prominent example of property (T) groups is SL(n,R). A group G is said to have property (T) if every affine isometric actions of G on a Hilbert space has a fixed point. Various fortifications of this property have been suggested by several researchers and proved for SL(n,R). In a series of lectures, I will talk about results in this direction of Lafforgue, Shalom, Mimura, and myself.
Abstract: To each dynamical system one can associate a space of cocycles (test functions) as well as a subspace of coboundaries, so that the associated cohomology reflects some of the underlying dynamics. In this series of lectures, we will deal with a generalization of this framework: the associated dynamics will correspond to that of a group action, and the cocycles will take values in the isometry group of a space of nonpositive curvature. As we will see, most of the classical theorems admit generalized versions in this setting (e.g. Birkhoff ergodic theorem, Gotsschalk-Hedlund theorem). Moreover, this gives a unified view with other classical results (e.g. Oseledets theorem). More importantly, this allows obtaining new results, as for instance: 1) The space of orientation-preserving C^1 actions of every nilpotent group on a 1-dimensional compact space os connected (N). 2) C^2 circle diffeomorphisms of irrational rotation number admit no invariant 1-distribution other than the invariant measure (N-Triestino). 3) Every linear cocycle can be perturbed so that to become conformal along the Oseledets splitting. In case all Lyapunov exponents are zero, then it can be perturber so that to become cohomologous to a cocycle of rotations (Bochi-N). Several open questions will be addressed.
Abstract: Given a group G acting on a probability space X by measure preserving transformations, one has a corresponding a unitary representation of G (Koopman representation); an important question, with diverse applications, is whether this action has a spectral gap, a rigidity property defined in terms of this representation. Actions of groups with property (T) always have such a spectral gap property. We will review some recent results as well as a few applications on this question in different cases: G is a subgroup of a Lie group H acting on X=H/L for a lattice L in H; G is a group of automorphisms of a torus or, more generally, of a nilmanifold X.
Abstract: A relatively hyperbolic group acts on a Gromov hyperbolic space. Its Gromov boundary has no information on peripheral subgroups since the boudnary of the orbit of any point by a peripheral subgroup consists only of one point, called cusp. We construct a blow up of cusp and give a nice boundary of a relatively hyperbolic group. We show that, under appropriate assumptions on peripheral subgroups, the $K$-homology of this boundary is isomorphic to the $K$-theory of the Roe-algebra of the group. As application, we give an explicit computation of the $K$-theory of the Roe-algebra of the fundamental group of the complement of a hyperbolic knot. If time permits, we will discuss a dual theory, that is, the $K$-theory of the stable Higson corona, coarse $K$-theory and coarse co-assembly map.
Abstract: In this talk, we consider coarse nonembeddability of a graph containing a sequence of expanders as induced subgraphs into Hilbert spaces. Using this, we see that a graph containing a "generalized" sequence of expanders (a sequence of finite graphs which have uniformly bounded k-th eigenvalues of the Laplacians and uniformly bounded degrees and whose numbers of vertices diverge to infinity) is not coarsely embeddable into Hilbert spaces.
Abstract: By applying concepts in the quasiconformal Teichm\"uller theory, we give a necessary and sufficient condition for a non-abelian group $G$ of $(1+\alpha)$-diffeomorphisms of the circle with $\alpha>1/2$ to be conjugated to a group of M\"obius transformations by a diffeomorphism in the same class. In its argument, we also see certain rigidity of such a group $G$ in the deformation given by conjugation of symmetric self-homeomorphisms of the circle.
Abstract: I will formulate a version of law of large number in the strong sense for random walks on Lie groups, and show this is useful to describe the boundary associated with a group and a random walk on it. More precisely, for example, for nilpotent Lie groups, we can show that the asymptotic direction of random walks on them is unique almost surely. But, even for some polycyclic groups, the asymptotic direction of random walks on them is not unique any more, due to Kaimanovich. I explain how the asymptotic direction describes the boundary. I will also give some questions around this construction. | CommonCrawl |
In the fundamental works on the theory of Lie groups (S. Lie, H. Poincaré, E. Cartan, H. Weyl, and others) it is a group of smooth or analytic transformations of the space $\mathbf R^n$ or $\mathbf C^n$, depending smoothly or analytically on parameters. When there are finitely many numerical parameters, a continuous group is called finite, which corresponds to the modern concept of a finite-dimensional Lie group. In the presence of parameters that are functions one speaks of an infinite continuous group, which corresponds to the modern concept of a pseudo-group of transformations. Nowadays (1988) the term "continuous group" often stands for topological group .
This page was last modified on 9 July 2014, at 23:21. | CommonCrawl |
A Dirichlet character $\chi: \Z\to \C$ has a modulus $n$ such that $\chi$ is induced from a function $\chi:\Z/n\Z\to \C$. It gives a homomorphism $(\Z/n\Z)^\times\to \C^\times$. The underlying unit group is $(\Z/n\Z)^\times$, the group of units of the ring $\Z/n\Z$.
Not referenced anywhere at the moment. | CommonCrawl |
"The P2NFFT method for mixed charge-dipole systems"
We consider a typical $N$-body problem in order to compute electrostatic interactions in particle systems containing a mixture of charges and dipoles. Classical particle-mesh methods make use of the fast Fourier transform (FFT) to compute the interactions in pure charge systems subject to periodic boundary conditions in all three spatial directions. This enables the approximation of the desired quantities with only $\mathcal O(N\log N)$ arithmetic operations, where $N$ denotes the number of present charges. Particle systems containing a set of dipoles have already been studied as well and may be treated in a similar fashion. One particle-mesh method is called the particle-particle NFFT (P$^2$NFFT), which is based on the nonuniform fast Fourier transform (NFFT). Recently, this method has been generalized to 2d-periodic, 1d-periodic as well as open boundary conditions. In addition, the approach has been extended for the treatment of particle systems containing a mixture of charges and dipoles. Consequently, we present for the first time an efficient $\mathcal O(N\log N)$ algorithm for mixed charge-dipole systems, that in addition allows the handling of various types of periodic boundary conditions based on a unified framework. The method is publicly available. Numerical results confirm that the method can be tuned to high accuracies. | CommonCrawl |
It looks very similar indeed, except for a rotation of each velocity by 90° (in the complex plane) and complex conjugation. Namely, the vortices tend to swirl around one another, not collide as the polynomial zeros tend to do under heat flow. In fact, the motion of vortices is described by a Hamiltonian dynamical system (for equal strength vortices is the generalized position and is the generalized momentum) where the conserved Hamiltonian is, for equal strength vortices, precisely the entropy defined for the heat flow . This immediately shows that point 2D vortices can in fact never collide.
We can get a bit closer to vortex motion if we Wick rotate (meaning we replace with , i.e., we evolve "in complex time") the heat Eq. (2) into the Schrödinger equation: . Then the Wick rotated solution for the polynomial zeroes Eq. (3) acquires an extra which is needed to rotate the velocity by 90° in the complex plane. The "only" thing that is missing is complex conjugation of the velocity (i.e. reflection across the real line).
This missing complex conjugation is quite important, however, as at least my numerical simulations tend to show that the polynomial zeroes now mostly repel one another and at late times quickly approach pair-wise diverging motion along the directions (I checked for up to 4th order polynomials), but do not seem to ever collide at least in generic cases (there are fine-tuned degenerate cases like the polynomial, though, where they do collide, and cases like polynomials of order 3 where a pair of zeroes diverges along but one zero remains near the origin). The behaviour is similar to dynamical-system motion around a hyperbolic point instead of an elliptical one as for the motion of point vortices.
At present I do not see a trivial deformation of the heat/Schrödinger equation that would give the motion of polynomial zeros that is the same as the motion of point vortices (complex conjugating P on one side of the heat/Schrödinger equation does not do it, of course). It is intriguing how close it seems, though.
yeah,i just do a similar calculate and get the same dynamic picture in my mind,but i do not think the whole story in a physics picture.anyway,one key obeserve is that the points should diverging along n distinct line to infinity in the complex plane .at least in the case the limit polynomial has n distinct root.for the degenerate situation the thing is not so easy to control we need to analysis to certain entanglement pairs.and if is odd,then we will have a root 0 in the limit case,this is also a annoying thing.
in my opinion,i think we need some knowledge from buried group to investigate a fix dynamic system generat by the zero of polynomial under heat flow,just think with the situation for quadratic polynomial ,this will corresponding to 2 different dynamic picture,because we do not know what happen at (0,0).this is a broken of symmetry.we need to use buried group and the jordan curve theorem(the topology only change when two point collide) to find the loss information.by the way,i think consider the deformation of zeros for polynomial function under heat equation on compact complex analytic surface(especially on ) is also interesting.
You appear to have confused the "fundamental theorem of arithmetic" with the "fundamental theorem of algebra" at the very beginning of this post.
I would like you to know that I am truly honored to have the opportunity to point this out to you. Great post otherwise and I really appreciate the fact that you put these posts out!
P.S. In case the tone is unclear over text, I'm not pointing this out in a demeaning manner or even out of serious concern. I simply never thought I'd ever catch one of your mistakes, and I found so ironic that THIS was the mistake, that I simply had to say something! Once again, I really appreciate the fact that you make these posts!
Geniuses make mistakes too (Though it is a unintentional) !
It is a typo, no confusion.
I believe there is a typo in the series expansion of $P(t,z)$. Apparently the The right-hand side of the equation just after (2) does not depend on $t$.
Instead of "1" on the RHS, it should be $t^n$.
If is (locally) analytic in , it has (locally) the same zero set as its Weierstrass polynomial (in – with analytic coefficients in ) – so the (local) dynamics of the zeros can be represented by their Puiseux series. Is it possible to extend (locally) the above results for (possibly infinitely many) zeros arranged in separated clusters of (possibly colliding) zeros?
Beginner Q: Is it implicit in (2) that also P(0,z) = P(z)? Otherwise (2) does not seem to establish a relationship between P(t,z) and P(z).
Treating it as an optimal transport problem, I wonder if the behavior of the zeroes is a consequence of the relationship between energy geodesics, entropy and the manifold curvature (as treated for other dynamics by, for example, Cedric Villani in his "lazy gas experiment").
Fundamental theorem of algebra of course, not arithmetic.
Dimensions are incorrect in the Taylor series for [math]P(t,z)[/math]. The right side needs [math]t^n[/math] inside the summation.
exactly,so we can rescaling it and take $t\to \infty$,the limit case is the equation: (*) this have n zero on .
until now,at least for the case the n zero is distinct in (*),we know at last the zero will go to infinity along each direction come form the zeros of (*),so the only complicated thing is the finite "blow up"time i.e. the time zeroes must collide.these will lead to to breaken of symmetry(just think about the example ,there is only one equation,but two different pictures).to investigate this I think we need some knowledge about buried group.
The letter n is used as both the degree of the polynomial and the summation index of the Taylor series. Perhaps a different letter could be used in the latter purpose to avoid any possible confusion.
Your equation (3) reminds one of the dynamics of zeros of a power series when the coefficients undergo Brownian motions. The difference from Dyson's BM is that the inverse distances to other zeros appears in the variance (not the drift) of the diffusion describing the motion of a particular root.
Does this have connection with Lee-Yang theorem about the zeros of partition function in statistical mechanics?
I am not an expert on the Lee-Yang theorem, but there is at least a small but interesting historical connection — see the (well written and self-contained) remarks made by Mark Kac at the end of the 3rd volume of Polya's collected works, on Polya's paper `Bemerkung uber die integraldarstellung der Riemannschen -funktion'.
Solutions of the form for all k give the equations of Stieltjes for the optimizer of the Coulomb gas model as in Mehta appendix 6, whose solution is Hermite zeros in random matrix theory. This is obvious b/c the Coulomb gas model is H plus the sum-of-squares. But it is also the self-similar solution of the dynamics.
Thanks for the reference. What is the motivation for the sum-of-squares term in that model?
Equation after "in time using (2) to obtain" should be a difference rather than a sum, no?
Does anyone have any insight on how one could come up with the sum-of-squares representation for in Exercise 1, if one did not know it in advance? How are these algebraic identities discovered by mathematicians in practice?
As I wrote in the post, this identity ultimately arises from the convexity of . If obeys (reverse) gradient flow , then . If is convex, then is positive definite and the RHS should be expressible as the sum of squares. If one pursues this line of reasoning a bit more, one will ultimately arrive at the identity in Exercise 1.
Oh, nice! Thanks so much for your reply. I should have paid more attention to the sentence about convexity in the post.
I was just interested in the following: Let's say we've got a random matrix. We can associate to it a monic polynomial by taking the characteristic polynomial, and then develop it with the heat equation (or any other equation). Then all the quantities above become random variables, dependent on the entries of the random matrix.
It seems that random matrices and heat flows inhabit parallel, but disconnected, "worlds", parameterised by an inverse temperature parameter , and that it is not particularly natural to mix the different worlds together. Heat flow corresponds to the deterministic ( ) world, being connected in particular to the finite free convolution of Marcus, Spielman, and Srivastava. The random matrices relating to the Gaussian Unitary Ensemble instead lie in the world, the random matrices relating to the Gaussian Orthogonal Ensemble lie in the world, and so forth. Each has its own flow; for it would be the Dyson Brownian motion, which looks nearly identical to the heat flow dynamics of zeroes but with an additional Brownian drift term, and corresponds to perturbing each entry of the matrix by a (complex) Gaussian perturbation (subject to maintaining the Hermitian property). | CommonCrawl |
We know that the consistency of ZFC+"Exists an inaccessible cardinal" implies the consistency of ZF+DC+"All sets are Lebesgue measurable"; and DC proves the existence of non-Borel sets.
J. Truss proved that repeating Solovay's construction by collapsing any limit cardinal to be $\aleph_1$ we obtain a model of ZF+"All sets are Lebesgue measurable", and in that model DC holds if and only if we collapsed an inaccessible. If we collapsed a singular cardinal then the resulting model has the property that all sets are Borel.
If we assume ZF+"All sets are Lebesgue measurable"+"There exists a non-Borel set", can we conclude that there is an inner model with an inaccessible cardinal?
When I say Borel sets, I mean elements of the $\sigma$-algebra generated by the open sets.
When I say Lebesgue sets, I mean elements of the $\sigma$-algebra generated by completing the Borel $\sigma$-algebra with respect to the null ideal.
Assume that ZF+"All sets are Lebesgue measurable"+"The Borel measure is $\sigma$-additive", can we conclude that there is an inner model with an inaccessible cardinal?
Now, taking the Borel and Lebesgue sets as defined above makes more sense.
If the Borel measure is not $\sigma$-additive, can we represent $\mathbb R$ as a countable union of null sets?
Browse other questions tagged set-theory lo.logic axiom-of-choice or ask your own question.
Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?
How strong is "all sets are Lebesgue Measurable" in weaker contexts than ZF?
Pathological behavior of Borel sets?
Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF? | CommonCrawl |
Journal: J. London Math. Soc.
We establish existence and non-existence results for entire solutions to the fractional Allen-Cahn equation in $\mathbb R^3$, which vanish on helicoids and are invariant under screw-motion. In addition, we prove that helicoids are surfaces with vanishing nonlocal mean curvature. | CommonCrawl |
Department of Mathematics, Shimane University, Matsue, Japan.
On the set $\mathbb R$ of real numbers we consider a poset $\mathcal P_\tau(\mathbb R)$ (by inclusion) of topologies $\tau(A)$, where $A\subseteq \mathbb R$, such that $A_1\supseteq A_2$ iff $\tau(A_1)\subseteq \tau(A_2)$. The poset has the minimal element $\tau (\mathbb R)$, the Euclidean topology, and the maximal element $\tau (\emptyset)$, the Sorgenfrey topology. We are interested when two topologies $\tau_1$ and $\tau_2$ (especially, for $\tau_2 = \tau(\emptyset)$) from the poset define homeomorphic spaces $(\mathbb R, \tau_1)$ and $(\mathbb R, \tau_2)$. In particular, we prove that for a closed subset $A$ of $\mathbb R$ the space $(\mathbb R, \tau(A))$ is homeomorphic to the Sorgenfrey line $(\mathbb R, \tau(\emptyset))$ iff $A$ is countable. We study also common properties of the spaces $(\mathbb R, \tau(A)), A\subseteq \mathbb R$. | CommonCrawl |
I was reading a paper in which the authors use the fact that any compact simply-connected homogeneous symplectic manifold has non-zero Euler characteristic. They prove it by quoting a theorem by Kostant which implies that the manifold is symplectomorphic to a coadjoint orbit of a semisimple group, then state that compact coadjoint orbits of semisimple groups have non-zero Euler characteristic.
I am looking for a more direct proof of that fact. Do you know some?
Let your manifold be $X=G/H$. First of all, since it is simply connected, we can write it as $K/U$ where $K$ and $U=K\cap H$ are compact in $G$ (Montgomery's theorem, 1950). Next, since $K/U$ is homogeneous symplectic, one knows that $U$ is the centralizer of a torus $S\subset K$.1) In particular $U$ contains any maximal torus containing $S$, i.e. $U$ is an equal rank subgroup of $K$. And finally, one knows that equal rank subgroups satisfy $χ(K/U)\ne0$: e.g. Samelson (1958), or Mostow (2005).
1) That is clear, with $S$ the closure of $\exp(\mathbf Rx)$, if we already know that $X\simeq$ the (co)adjoint orbit of some $x\in\mathfrak k^*\simeq\mathfrak k$. But it can also be proved a priori : Borel–Weil (1954, Thm 1), or in more detail Matsushima (1957, Thm 1).
Not the answer you're looking for? Browse other questions tagged sg.symplectic-geometry symplectic-topology homogeneous-spaces euler-characteristics or ask your own question.
Test for Homogeneity of Symplectic Manifolds?
When is a coadjoint orbit an integrable system (in a weak sense explained below)?
Cotangent bundle of coadjoint orbit is stein manifold?
When does a symmetric Poisson manifold decompose into homogeneous pieces?
Lagrangian homology classes in compact symplectic manifolds? | CommonCrawl |
I am trying to figure out why the answer is T>W. Do you have any idea why? If so could you please explain. Thank you.
There is an explanation in the diagram. What don't you like about this explanation?
So the denominator $2\sin\theta\cong0$ and therefore $T\to\infty$. As $W=mg$, clearly $W$ is a finite value (making the reasonable assumption that the mass is not inifite), and therefore clearly $T>W$.
This answer is already provided in the question. | CommonCrawl |
For puzzles based on the movements of a knight piece in chess.
On a $3\times3$ grid we have: with $8$ moves needed to swap the red and blue knights. What is the minimum numbers of moves to swap the knights on a $4\times4$ grid?
A knight is placed on an infinitely large chess board with no edges. It can only visit each square once. What is the smallest number of moves it can make that would cause it to become trapped? | CommonCrawl |
Let $R_1,\ldots,R_n$ and $C_1,\ldots,C_n$ be sets of size n.
When does there exist an $n \times n$ matrix in which the $i$-th row is a permutation of $R_i$, for all $1 \leq i \leq n$, and the $j$-th column is a permutation of $C_j$, for all $1 \leq j \leq n$?
The multiset $\cup R_i$ equals the multiset $\cup C_i$.
where $A$ and $B$ are $n \times n$ matrices, and $\emptyset$ represents $n \times n$ empty blocks. When does such a partial Latin square complete?
Browse other questions tagged co.combinatorics latin-square or ask your own question.
How many n×n (0,1)-matrices with row/column sum 4 and trace 0 are there?
Is a "row discrepancy" of symmetric row-column increasing matrices unbounded?
Proving that the set of $\lfloor n/3 \rfloor+1$ partial Latin squares given by Pebody is unavoidable?
Counting 2m X 2m 0-1 matrices with m ones in each row and each column.
For which divisors $a$ and $b$ of $n$ does there exist a Latin square of order $n$ that can be partitioned into $a \times b$ subrectangles?
Is there a way to estimate the number of Latin squares with a given autotopism?
Has the existence of a 3-MOLS(10) containing a self-orthogonal Latin square and its transpose been eliminated? | CommonCrawl |
In this article, I will talk about how to write Monte Carlo simulations in CUDA. More specifically, I will explain how to carry it out step-by -step while writing the code for pricing a down-and-out barrier option, as its path dependency will make it a perfect example for us to learn Monte Carlo in CUDA. Also, I will show you how to efficiently generate random numbers with CUDA and how to measure performance with just a few lines of code.
First, I will start with a brief theoretical introduction so if you already know how Monte Carlo methods and barrier options work, you can skip the following sections.
A barrier option is an exotic derivative, part of the set of path-dependent options, whose payoff depends not only on the underlying price at maturity but also on whether the price line hit a pre-determined level.
There are different ways to determine this level and how the price can or cannot reach it. The first is the barrier level position in relation to the current underlying price (spot), so we have a first categorization "up" or "down". The second criterion can be "in" or "out", and it refers to what happens when the event "hit the level" is triggered.
"In" means that the option starts to be active after having touched the barrier level. "Out" is the opposite, meaning that after triggering the event the option is no longer valid. Also, it could be "paired" with any kind of option. We could have a European option with barrier, as well as an American or an Asian one. Let's consider an example now.
The underlying price hits the barrier before the maturity making it invalid. Sometimes, as a kind of insurance, we can have a rebate price, which is a fixed amount of money, usually less than the option value, which we will receive in case our option expires due to hitting the barrier. Of course, this will also change the price of the option itself.
where $R$ is the rebate price and $B$ the barrier level.
The Monte Carlo method is a well-known method in finance, as it lets us compute difficult, if not impossible, expected values of complex stochastic functions. Mike has already discussed the method in several articles regarding option pricing, but a few recap lines can be helpful for those that are new to it.
The Monte Carlo method was first introduced in the field of physics, for complex simulations, very likely by Enrico Fermi in the 1930s for studying neutron diffusion. It then became popular in the 1940s among physicists and mathematicians involved in creating bombs for the U.S. Army. The projects needed a code name, so John Von Neumann chose "Monte Carlo", referring to the famous Monte Carlo Casino. Since then, technology and especially computational power have increased dramatically, letting us use these methods for a large variety of problems.
In finance the Monte Carlo method is mainly used for option pricing as, especially with exotic options, the payoff is sometimes too complex, if not impossible, to compute. The main idea behind it is quite simple: simulate the stochastic components in a formula and then average the results, leading to the expected value. Of course, the more simulations (paths) you make, the more accurate the result will be. A commonly accepted value for the minimum number of paths is $10^6$. That should give good results for most of the simulations. Otherwise, there are techniques that can reduce variance in order to make even more accurate predictions.
Given the random nature of this process, variance reduction is not the only problem we can encounter. Another one, probably the most important, is how the random numbers are generated. There is an entire branch of mathematics talking about this and a detailed explanation is well beyond the purpose of this article, but we will see that CUDA can provide different efficient methods for generating random numbers by including the useful library curand.
As the Monte Carlo method is basically a way to compute expected values by generating random scenarios and then averaging them, it is actually very efficient to parallelise. Moreover, with consumer CPUs on standard computers it is just not possible to reach the accuracy needed, as simulating over one million paths is usually very time consuming. With the GPU we can reduce this problem by parallelising the paths. That is, we can assign each path to a single thread, simulating thousands of them in parallel, with massive savings in computational power and time. At the end of this article I will show you the numerical results, making it quite obvious why it's better to run a Monte Carlo on a GPU.
First, let's see what and how to parallelise. In option pricing, usually the only variable that can assume random values is the underlying, so we only have to write a kernel that can generate a simulated value for the underlying and then calculate the option price. That's it. Sounds easy, but actually we have to cope with a couple of issues that we could have avoided for the pricing of a path-independent option.
That is, as the barrier can be hit at any point in time we have to simulate step by step the changes in the underlying price, significantly reducing the code speed. Why? Let's say that we want to run an accurate Monte Carlo, which means more than one million paths. And let's say that we also want to use a reasonable proxy for price changes, i.e. only simulate daily changes. This means that we will have to generate 365*10^6 random numbers as well as perform 365 price computations one million times! Also, for more complex derivatives or for purposes other than learning, daily changes will likely not have sufficient granularity.
Before having a look at the code, let me give you the last theoretical basis you need (if you don't know it already) to fully understand this method.
For simulating the underlying price, we must discretise the underlying's changes. In this article, I will make use of the Euler method, as it's very easy to understand (and code up) and, despite the fact that it isn't the best method, it's still a good approximation for our needs.
$Y$ is the price at the time step $n$, where $Y_0 = S_0$.
Now we have to compute the changes. As you can see from the list above, the only random variable is $dW$. This variable is the only reason why we need to run a Monte Carlo simulation.
Now we are ready to have a look at the code.
You can get the number of clock ticks elapsed since the program started. This is a C function and, as it can be affected by many factors, it's better to never use it alone. Instead you can compute the difference between two times and then get the time (in seconds) of a given task or code portion.
This simple code will give you the exact time elapsed for the // routine part of the code. CUDA also provides a library for this purpose, but for now the C one is more than sufficient for us.
CUDA provides efficient random number generators for a lot of different distributions via the library curand.h. In this case, as the Brownian motion evolves with normally distributed random steps, we will use the normal generator.
This first one is just a variable declaration, in which we are creating the new generator as a variable of type curandGenerator_t, called curandGenerator.
Now it's time to finally generate our normally distributed random numbers. We can do this by using the curand function curandGenerateNormal, which takes as inputs the curand generator, the output array (in which we want to store the numbers), the amount of numbers to generate, the mean of the distribution and its standard deviation. In this case, as we are talking about Brownian motion, we will need a normal distribution with mean 0 and variance $dt$.
Now let's talk about the main part, looking at the code. At the end of this article you will find the complete code, so now I will explain it step by step.
This first part consists of including libraries and variable declaration, but it is useful to notice a few choices I made. First, the try instruction: this is an additional error checking line, as if we have any problem, the program won't crash but will return the error information (using the instruction catch at the end of the code). This is good practice for longer programs and therefore a good habit to develop.
Regarding the parameters, you can see that I divided them into different blocks, reflecting their differing nature. In the first one we can find the "dimensional constants", or rather the lengths of our arrays and loops. N_PATHS specifies the number of paths (or runs) that the Monte Carlo method will perform. In this case we have $5.0 \times 10^6$, which is a reasonable number for having a good precision in the estimation. Then, as aforementioned, I decided to compute daily changes, setting N_STEPS = 365. Therefore the number of normals we will need is N_PATHS * N_STEPS, as we will need 365 random changes for $5.0 \times 10^6$ simulations.
That is a huge constraint for our precision, as this big array will have to be allocated in the GPU memory. So, we can choose to increase the precision of a single run (by increasing N_STEPS) or the overall accuracy (by increasing N_PATHS), until reaching the size limit for device allocation, which depends solely on your GPU. In this case I decided that 365 was a reasonable approximation, then I maximized N_PATHS, but feel free to experiment, as usually is the best way to learn!
The second and the third blocks represent our input parameters. More specifically, the second is for constant declaration, in which constants are "market parameters" necessary for computing the option price, while the third block is for derived variables.
The fourth is for array declarations. s is the host array that receives the final prices after they will be computed by the GPU, d_s is exactly the same array but for the device (GPU), and the last one is the array that contains the random numbers. d_s and d_normals are declared using the class dev_array.h that I showed in my previous article. So, if you haven't read it already, it might be worth having a look at it, as it will be used several times for this script.
I decided to use 1024 threads per block, even if I didn't notice significant changes in the performance between setting BLOCK_SIZE=1024 and BLOCK_SIZE=256 but, as we will have a lot of threads working (one for each path!), 1024 is a reasonable choice.
So that now we are ready to start with the actual Monte Carlo loop.
Second, we have to write a do/while loop. That's a bit different from the usual Monte Carlo methods, which make use of a normal for loop but it reflects the path-dependent nature of the barrier option. In fact if we hit the barrier our down-and-out option will no longer be active, so continuing to simulate that path to maturity wouldn't make sense anymore.
You can see here there are two conditions for staying in the loop: the first one is that the number of steps already made is lower than the maximum number of steps (365 in this case) allowed, while the second one is that the current price is still higher than the barrier.
We then update the price making use of the Euler discretisation and after that we update our indexes. That is, we first update the normal array index n_idx and then the loop index n, that states which "day" the loop is computing.
Using only 365 steps we are missing the cases in which the price fell under the barrier during a certain day and then closed higher than the barrier at the end of the day: this reduces the accuracy of the price but, as I already said, it's part of the trade-off between the accuracy of the expected value and the simulated price path, with your GPU capability acting as your only constraint.
Which is exactly the payoff of a plain vanilla call.
Now we have to compute the expected value, averaging all the prices that we got from the kernel.
First, we need to synchronize the device and to copy the prices from the device to the array.
The first is the usual CUDA standard method for synchronizing, while d_s.get() is the dev_array function for copying data from device to host.
What follows is the for loop for computing the price sum and, thus, our expected price value.
Now we have the price of a down-and-out barrier option in CUDA computed via the Monte Carlo method. Notice that it can also compute a European call just by setting the barrier value to 0.0f!
We can see that the GPU implementation was roughly 537x faster than the CPU one, including the memory allocation host to device.
In future articles we will also talk about exploiting CUDA using different pricing methods, including multidimensional finite differences methods. | CommonCrawl |
To find the trajectory of anything in General Relativity, usually you only need the metric tensor, from which you can obtain the geodesic equations. Nevertheless, a common problem that arises in cosmology is that as soon as we depart from the simplest homogeneous models, the task of finding solutions to the geodesic equations quickly becomes an intractable analytical problem.
In this post are some notes of how to perform numerical integration of light paths in the Schwarzschild metric.
We are interested in the trajectory of a light ray in such a metric. Since the metric is spherically symmetric, any light ray that starts with a certain $\theta$ must stay in the same $\theta$ plane, hence we can arbitrarily set $\theta = \pi/2$ and do away with all the $\theta$ terms.
where an overdot refers to derivative with respect to an affine parameter $\lambda$.
In principle, we need the initial values of $r$, $p$, and $\phi$ to start the numerical simulation. However, if we fix the incoming velocity to be horizontal, then we would only need to specify the initial $x_0$ and $y_0$ coordinates.
Then, the only free parameters to specify $b$ and $x_0$, in addition to mass. | CommonCrawl |
Multivariate clustering in astrophysics is a recent development justified by the bigger and bigger surveys of the sky. The phylogenetic approach is probably the most unexpected technique that has appeared for the unsupervised classification of galaxies, stellar populations or globular clusters. On one side, this is a somewhat natural way of classifying astrophysical entities which are all evolving objects. On the other side, several conceptual and practical difficulties arize, such as the hierarchical representation of the astrophysical diversity, the continuous nature of the parameters, and the adequation of the result to the usual practice for the physical interpretation. Most of these have now been solved through the studies of limited samples of stellar clusters and galaxies. Up to now, only the Maximum Parsimony (cladistics) has been used since it is the simplest and most general phylogenetic technique. Probabilistic and network approaches are obvious extensions that should be explored in the future.
Observations of astrophysical objects such as galaxies are limited by various sources of random and systematic noise from the sky background, the optical system of the telescope and the detector used to record the data. Conventional deconvolution techniques are limited in their ability to recover features in imaging data by the Shannon-Nyquist sampling theorem. Here we train a generative adversarial network (GAN) on a sample of $4,550$ images of nearby galaxies at $0.01<z<0.02$ from the Sloan Digital Sky Survey and conduct $10\times$ cross validation to evaluate the results. We present a method using a GAN trained on galaxy images that can recover features from artificially degraded images with worse seeing and higher noise than the original with a performance which far exceeds simple deconvolution. The ability to better recover detailed features such as galaxy morphology from low-signal-to-noise and low angular resolution imaging data significantly increases our ability to study existing data sets of astrophysical objects as well as future observations with observatories such as the Large Synoptic Sky Telescope (LSST) and the Hubble and James Webb space telescopes.
K. Schawinski, C. Zhang, H. Zhang, et. al.
The inference of correlated signal fields with unknown correlation structures is of high scientific and technological relevance, but poses significant conceptual and numerical challenges. To address these, we develop the correlated signal inference (CSI) algorithm within information field theory (IFT) and discuss its numerical implementation. To this end, we introduce the free energy exploration (FrEE) strategy for numerical information field theory (NIFTy) applications. The FrEE strategy is to let the mathematical structure of the inference problem determine the dynamics of the numerical solver. FrEE uses the Gibbs free energy formalism for all involved unknown fields and correlation structures without marginalization of nuisance quantities. It thereby avoids the complexity marginalization often impose to IFT equations. FrEE simultaneously solves for the mean and the uncertainties of signal, nuisance, and auxiliary fields, while exploiting any analytically calculable quantity. Finally, FrEE uses a problem specific and self-tuning exploration strategy to swiftly identify the optimal field estimates as well as their uncertainty maps. For all estimated fields, properly weighted posterior samples drawn from their exact, fully non-Gaussian distributions can be generated. Here, we develop the FrEE strategies for the CSI of a normal, a log-normal, and a Poisson log-normal IFT signal inference problem and demonstrate their performances via their NIFTy implementations.
State of the art methods in astronomical image reconstruction rely on the resolution of a regularized or constrained optimization problem. Solving this problem can be computationally intensive and usually leads to a quadratic or at least superlinear complexity w.r.t. the number of pixels in the image. We investigate in this work the use of convolutional neural networks for image reconstruction in astronomy. With neural networks, the computationally intensive tasks is the training step, but the prediction step has a fixed complexity per pixel, i.e. a linear complexity. Numerical experiments show that our approach is both computationally efficient and competitive with other state of the art methods in addition to being interpretable.
Celeste is a procedure for inferring astronomical catalogs that attains state-of-the-art scientific results. To date, Celeste has been scaled to at most hundreds of megabytes of astronomical images: Bayesian posterior inference is notoriously demanding computationally. In this paper, we report on a scalable, parallel version of Celeste, suitable for learning catalogs from modern large-scale astronomical datasets. Our algorithmic innovations include a fast numerical optimization routine for Bayesian posterior inference and a statistically efficient scheme for decomposing astronomical optimization problems into subproblems.
Our scalable implementation is written entirely in Julia, a new high-level dynamic programming language designed for scientific and numerical computing. We use Julia's high-level constructs for shared and distributed memory parallelism, and demonstrate effective load balancing and efficient scaling on up to 8192 Xeon cores on the NERSC Cori supercomputer.
J. Regier, K. Pamnany, R. Giordano, et. al.
Understanding the nature of dark energy, the mysterious force driving the accelerated expansion of the Universe, is a major challenge of modern cosmology. The next generation of cosmological surveys, specifically designed to address this issue, rely on accurate measurements of the apparent shapes of distant galaxies. However, shape measurement methods suffer from various unavoidable biases and therefore will rely on a precise calibration to meet the accuracy requirements of the science analysis. This calibration process remains an open challenge as it requires large sets of high quality galaxy images. To this end, we study the application of deep conditional generative models in generating realistic galaxy images. In particular we consider variations on conditional variational autoencoder and introduce a new adversarial objective for training of conditional generative networks. Our results suggest a reliable alternative to the acquisition of expensive high quality observations for generating the calibration data needed by the next generation of cosmological surveys.
S. Ravanbakhsh, F. Lanusse, R. Mandelbaum, et. al.
We apply a novel spectral graph technique, that of locally-biased semi-supervised eigenvectors, to study the diversity of galaxies. This technique permits us to characterize empirically the natural variations in observed spectra data, and we illustrate how this approach can be used in an exploratory manner to highlight both large-scale global as well as small-scale local structure in Sloan Digital Sky Survey (SDSS) data. We use this method in a way that simultaneously takes into account the measurements of spectral lines as well as the continuum shape. Unlike Principal Component Analysis, this method does not assume that the Euclidean distance between galaxy spectra is a good global measure of similarity between all spectra, but instead it only assumes that local difference information between similar spectra is reliable. Moreover, unlike other nonlinear dimensionality methods, this method can be used to characterize very finely both small-scale local as well as large-scale global properties of realistic noisy data. The power of the method is demonstrated on the SDSS Main Galaxy Sample by illustrating that the derived embeddings of spectra carry an unprecedented amount of information. By using a straightforward global or unsupervised variant, we observe that the main features correlate strongly with star formation rate and that they clearly separate active galactic nuclei. Computed parameters of the method can be used to describe line strengths and their interdependencies. By using a locally-biased or semi-supervised variant, we are able to focus on typical variations around specific objects of astronomical interest. We present several examples illustrating that this approach can enable new discoveries in the data as well as a detailed understanding of very fine local structure that would otherwise be overwhelmed by large-scale noise and global trends in the data. | CommonCrawl |
Suppose Ivica's message consists of $N$ characters.
Ivica must first find a matrix consisting of $R$ rows and $C$ columns such that $R \le C$ and $R \cdot C = N$. If there is more than one such matrix, Ivica chooses the one with the most rows.
Ivica writes his message into the matrix in row-major order. In other words, he writes the first segment of the message into the first row, the second segment into the second row and so on.
The message he sends to Marica is the matrix read in column-major order.
For instance, suppose Ivica wants to send the message "bombonisuuladici" containing 16 letters. He can use a $1 \times 16$, $2 \times 8$, or $4 \times 4$ matrix. Of these, the $4 \times 4$ has the most rows. When the message is written into it, the matrix looks like this, and the encrypted message becomes "boudonuimilcbsai".
Marica has grown tired of spending her precious time deciphering Ivica's messages, so you must write a program to do it for her.
The input contains the received message, a string of lowercase letters of the English alphabet (with no spaces). The number of letters will be between 1 and 100.
Output the original (decrypted) message. | CommonCrawl |
The Hodge-Deligne polynomials of some moduli spaces of coherent systems.
A coherent system on a smooth projective curve C consists of a pair (E,V) where E is a vector bundle on C (of rank n and degree d) and V is a subspace (of dimension k) of $H^0(C,E)$. For each triple (n,d,k) there is a family of moduli spaces of coherent systems, depending on a real positive parameter $\alpha$. It is known that these moduli spaces change only if we pass through a finite set of critical values, so we have a finite number of distinct moduli spaces labeled according to the corresponding interval in the real line. The final moduli space is in general very simple to study, while not so much is known about the intermediate moduli spaces and the first one (which has strong relations with the Brill-Noether locus $B(n,d,k)$). In particular, an interesting open problem is that of computing the Hodge-Deligne polynomials of such moduli spaces. I will present some explicit results in the cases (n=2,k=1) and (n=3,k=1), together with some general techniques that in principle could be used to tackle also more complicated cases. I will discuss also some partial results on the cases (n=4,k=1) and (n=2,k=2). | CommonCrawl |
Lets first define a few terms.
Discrete Random Variable - This is a variable that can only take on particular values. For example heads or tails of a coin, it makes no sense to talk about 3/4's heads or 1/8 tails. Another example, the integers 1 through 6 are possible values when rolling a die. There is no way to get 4.3 on a standard die.
Continuous Random Variable - This is in contrast to the definition above only in that the variable can take on any value in a range however, the range is generally limited by the context of the problem. For example, the height of human can take on virtually any value within a range (say 1 meter to 2.5 meters). It would be very odd to find that every person's height was an integer value of centimeters!
The table below shows the probability P(X) of seeing x number of heads after flipping a coin 4 times.
One way to visualize a random variable is to create a graph of all the possible values that the variable can take on. For a discrete variable this could look like a bar graph, with the height of the bar indicating the probability of that value. The bar graph below shows the probability of rolling each number on standard six-sided die.
Another example would be the probability distribution of flipping a coin twice. We can express the probability in terms of the number of heads that would be seen. Since there are four possible outcomes (HH, HT, TH and TT) each with equal probability (I'll let you work that out). The probability distribution look like the one below.
Where n is the number of possible outcomes, $x_i$ is the value of an outcome and $p_i$ is the probability that $x_i$ occurs.
This is a tough concept to explain (I didn't do a brilliant job) and maybe harder to understand. The expectation value is a way of giving a "weight" to each possible outcome. Lets use the example of rolling a six-sided die. Each outcome has an equal probability, but if I asked you to place a bet or make a guess about the most likely value that would occur what would you say…?
This can be interpreted as the value of each outcome ($x_i)$ multiplied by the probability ($p_i$) of that outcome. Each term in the summation is $x_i p_i$.
The IB likes the notation of $E(X)$ for the expectation value.
Dan Meyer has a nice post about expectation value. If you can follow what he's getting at your probably doing well.
Want to add to or make a comment on these notes? Do it below. | CommonCrawl |
This paper proposes the novel Pose Guided Person Generation Network (PG$^2$) that allows to synthesize person images in arbitrary poses, based on an image of that person and a novel pose. Our generation framework PG$^2$ utilizes the pose information explicitly and consists of two key stages: pose integration and image refinement. In the first stage the condition image and the target pose are fed into a U-Net-like network to generate an initial but coarse image of the person with the target pose. The second stage then refines the initial and blurry result by training a U-Net-like generator in an adversarial way. Extensive experimental results on both 128$\times$64 re-identification images and 256$\times$256 fashion photos show that our model generates high-quality person images with convincing details. | CommonCrawl |
Let $T$ be a random variable, and let $S$ be a random variable defined on the same space as $T$. As we have seen, conditioning on $S$ might be a good way to find probabilities for $T$ if $S$ and $T$ are related. In this section we will see that conditioning on $S$ can also be a good way to find the expectation of $T$.
We will start with a simple example to illustrate the ideas. Let the joint distribution of $T$ and $S$ be as in the table below.
How can $S$ be involved in the calculation of $E(T)$?
This is equivalent to going to each cell of the table, weighting the value of $T$ in that cell with the probability in the cell, and then adding. Here's another way of looking at this.
Each of the three conditional distributions is a distribution in its own right. Therefore its histogram has a balance point, just as the marginal distribution of $T$ does.
This defines a function of $S$: for each value $s$ of $S$, the function returns $E(T \mid S=s)$.
This function of $S$ is called the conditional expectation of $T$ given $S$ and is denoted $E(T \mid S)$. Unlike expectation which is a number, conditional expectation is a random variable.
As it's a random variable, it has an expectation, which we can calculate using the non-linear function rule. The answer is a quantity that you will recognize.
That's right: it's the expectation of $T$.
What we have learned from this is that $E(T)$ is the average of the conditional expectations of $T$ given the different values of $S$, weighted by the probabilities of those values.
In short, $E(T)$ is the expectation of the conditional expectation of $T$ given $S$.
In general, suppose $T$ and $S$ are two random variables on a probability space.
Then for each fixed value of $s$, $T$ has a conditional distribution given $S=s$. This is an ordinary distribution and has an expectation. That is called the conditional expectation of $T$ given $S=s$ and is denoted $E(T \mid S = s)$.
So for each $s$, there is a value $E(T \mid S=s)$. This defines a function of the random variable $S$. It is called the conditional expectation of $T$ given $S$, and is denoted $E(T \mid S)$.
$E(T)$, the expectation of $T$, is a real number.
$E(T \mid S)$, the conditional expectation of $T$ given $S$, is a function of $S$ and hence is a random variable.
Since $E(T \mid S)$ is a random variable, it has an expectation. That expectation is equal to $E(T)$. We observed this in an example; now here is a proof.
Suppose we want the expectation of a random variable, and suppose it is easy for us to say what that expectation would be if we were given the value of a related random variable. The rule of iterated expectations says that we can find that conditional expectation first, and take its expectation to get our answer.
Formally, let $S$ and $T$ be two random variables on the same space. Then $E(T) = E(E(T \mid S))$.
Let $X_1, X_2, \ldots $ be i.i.d. and let $E(X_1) = \mu_X$. Let $N$ be a non-negative integer valued random variable that is independent of the sequence of $X$'s and let $E(N) = \mu_N$.
where $S = 0$ if $N=0$.
Notice that $S$ is the sum of a random number of terms.
Answer. If $N$ were the constant 10, then the answer would be $10\mu_X$. This is our signal to condition on $N$. Here are the steps to follow.
This is an equality of real numbers. Note that it is true for all $n$, including 0.
Next write the conditional expectation in random variable notation.
This is an equality of random variables.
This is a natural answer. It is the expected number of terms being added times the expected size of each of those terms.
This is an important point to note about calculating expectations by conditioning. The natural answer is often correct.
In a Galton-Watson branching process, each individual has a random number of progeny. Assume that the numbers of progeny of the different indviduals are i.i.d. with mean $\mu$. Suppose the process starts with one individual in Generation 0.
Question. Assuming that there are no deaths, what is the expected total number of individuals in Generations 0 through $n$?
The value of $\mu$, the expected number of progeny of a single individual, determines how this expected total behaves as $n$ gets large. Even with no deaths, if $\mu < 1$ the expected population size tends to a positive constant as $n \to \infty$. But if $\mu \ge 1$ then the expected population size explodes.
The most important property of conditional expectation is the iteration that we have studied in this section. But conditional expectation has other properties that are analogous to those of expectation. They are now expressed as equalities of random variables instead of equalities of real numbers.
Go through the list and notice that all the moves you'd naturally want to make are justified. The proofs are routine; we won't go through them.
Two more properties formalize the idea that the variable that is given can be treated as a constant in conditional expectations.
"Constant": Let $g$ be a function. Then $E(g(S) \mid S) = g(S)$.
"Pulling out a Constant": $~E(g(S)T \mid S) = g(S)E(T \mid S)$.
though we sincerely hope you won't encounter a random variable as bizarre as this. | CommonCrawl |
Kindly organised by Jonas Azzam, this short mini course will be an insight into Cheeger's Theorem in Geometric Measure Theory.
Venue: 5.20 ICMS Lecture Theatre, 5th Floor, 47 Potterow, Edinburgh EH8 9BT.
Abstract: Rademacher's theorem states that any Lipschitz $f \colon \mathbb R^n \to \mathbb R$ is differentiable Lebesgue almost everywhere. It is a fundamental result in geometric measure theory. In 1999 Cheeger gave a very deep generalisation of Rademacher's theorem which replaces the domain with a doubling metric measure space that satisfies a Poincaré inequality.
This minicourse will give an overview of a new proof of Cheeger's theorem which uses modern techniques in the field of analysis on metric spaces. These techniques consider a rich structure of Lipschitz curves in the metric space, known as an "Alberti representation", which allow us to form a partial derivative of any Lipschitz function. By considering many such families of curves, we are able to form a derivative and hence deduce Cheeger's theorem. | CommonCrawl |
We have implemented a certified Wang tiling program for tiling a rectangle region using a brick corner Wang tile set. A brick corner Wang tile set is a special Wang tile set introduced by A. Derouet-Jourdan et al. in computer graphics in 2015 to model wall patterns texture. We have implemented a tiling algorithm using Coq proof assistant and proved its correctness. This correctness assures the existence of a tiling of any brick corner Wang tile set for any size of rectangle. The essential points of our proof are the existence of a tiling for a $2 \times 2$ rectangle and a simple induction process. Since the brick corner Wang tile is a class of infinite kinds of tile sets, it is not straightforward and there are many conditional branches to prove the correctness. The certification with Coq assures that there are no lack of conditions. | CommonCrawl |
Abstract: In this paper we show how the complexity of performing nearest neighbor (NNS) search on a metric space is related to the expansion of the metric space. Given a metric space we look at the graph obtained by connecting every pair of points within a certain distance $r$ . We then look at various notions of expansion in this graph relating them to the cell probe complexity of NNS for randomized and deterministic, exact and approximate algorithms. For example if the graph has node expansion $\Phi$ then we show that any deterministic $t$-probe data structure for $n$ points must use space $S$ where $(St/n)^t > \Phi$. We show similar results for randomized algorithms as well. These relationships can be used to derive most of the known lower bounds in the well known metric spaces such as $l_1$, $l_2$, $l_\infty$ by simply computing their expansion. In the process, we strengthen and generalize our previous results (FOCS 2008). Additionally, we unify the approach in that work and the communication complexity based approach. Our work reduces the problem of proving cell probe lower bounds of near neighbor search to computing the appropriate expansion parameter. In our results, as in all previous results, the dependence on $t$ is weak; that is, the bound drops exponentially in $t$. We show a much stronger (tight) time-space tradeoff for the class of dynamic low contention data structures. These are data structures that supports updates in the data set and that do not look up any single cell too often. | CommonCrawl |
whether the project is a part of a broader collaboration.
Application deadline for the year 2019 is December 31, 2018.
If you have a project that you believe is eligible for support from Votruba-Blokhintsev program, please submit your details below.
The application will be considered at the next meeting of Votruba-Blokhintsev Evaluation Committee and you will be notified about the result of evaluation process.
In the following fields, please feel free to use standard LaTeX symbols such as $\alpha$ (do not use any nonstandard ones).
One Czech and one BLTP JINR scientist has to be indicated who share a common responsibility for the grant management.
Before submitting please check carefully all above fields. | CommonCrawl |
Abstract: Let $\Omega$ be a convex domain in the complex plane $\mathbb C$ and $H$ the space of holomorphic functions in $\Omega$ with the topology of uniform convergence on compact subsets of $\Omega$. Let $W_1$ and $W_2$ be a pair of (differentiation) invariant subspaces of $H$ admitting spectral synthesis. Conditions ensuring that the intersection $W_1\cap W_2$ also admits spectral synthesis are described. One consequence of these conditions is a recent result of Abuzyarova (in a new, constructive and quantitative setting) on the representation of an invariant subspace admitting spectral synthesis as the solution space of a system of two homogeneous convolution equations.
New approximation results for entire functions of exponential type are used. | CommonCrawl |
Citation: Quantum 2, 81 (2018).
We provide a fine-grained definition for monogamous measure of entanglement that does not invoke any particular monogamy relation. Our definition is given in terms an equality, as oppose to inequality, that we call the "disentangling condition". We relate our definition to the more traditional one, by showing that it generates standard monogamy relations. We then show that all quantum Markov states satisfy the disentangling condition for any entanglement monotone. In addition, we demonstrate that entanglement monotones that are given in terms of a convex roof extension are monogamous if they are monogamous on pure states, and show that for any quantum state that satisfies the disentangling condition, its entanglement of formation equals the entanglement of assistance. We characterize all bipartite mixed states with this property, and use it to show that the G-concurrence is monogamous. In the case of two qubits, we show that the equality between entanglement of formation and assistance holds if and only if the state is a rank 2 bipartite state that can be expressed as the marginal of a pure 3-qubit state in the W class.
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The continuty of the convex roof extended entanglement measure can be checked according to Proposition 2 in supGuo2013qip, the continuty of partial trace and partial transpose is proved in supGuo2013csb, the continuty of the realtive entropy entanglement is proved in supDonald1999pla.
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Yu Guo, "Any entanglement of assistance is polygamous", Quantum Information Processing 17 9, 222 (2018).
Yu Guo and Gilad Gour, "Monogamy of the entanglement of formation", Physical Review A 99 4, 042305 (2019).
Christopher Eltschka, Felix Huber, Otfried Gühne, and Jens Siewert, "Exponentially many entanglement and correlation constraints for multipartite quantum states", Physical Review A 98 5, 052317 (2018).
Yu Guo, "Strict entanglement monotonicity under local operations and classical communication", Physical Review A 99 2, 022338 (2019).
Zhi-Xiang Jin and Shao-Ming Fei, "Polygamy relations of multipartite entanglement beyond qubits", Journal of Physics A: Mathematical and Theoretical 52 16, 165303 (2019).
Zhi-Xiang Jin, Shao-Ming Fei, and Xianqing Li-Jost, "Generalized Entanglement Monogamy and Polygamy Relations for N-Qubit Systems", International Journal of Theoretical Physics 58 5, 1576 (2019).
Soorya Rethinasamy, Saptarshi Roy, Titas Chanda, Aditi Sen(De), and Ujjwal Sen, "Universality in distribution of monogamy scores for random multiqubit pure states", Physical Review A 99 4, 042302 (2019).
Zhi-Xiang Jin and Shao-Ming Fei, "Superactivation of monogamy relations for nonadditive quantum correlation measures", Physical Review A 99 3, 032343 (2019).
The above citations are from Crossref's cited-by service (last updated 2019-04-25 17:09:23) and SAO/NASA ADS (last updated 2019-04-25 17:09:23). The list may be incomplete as not all publishers provide suitable and complete citation data. | CommonCrawl |
Here is the problem which I thought was simple dynamic programming, which is however not the case.
Given an $N \times M$ matrix of numbers from 1 to $NM$ (each number occurs only once), find a path from top left to right bottom while moving right or down only. If we sort all values visited in this path it should be lexicographically smallest.
I thought smallest sum path will be the answer, but it need not be true.
To implement this algorithm efficiently, we need an efficient data structure for the two-dimensional range minimum query problem. Brodal, Davoodi and Rao give, in their paper On Space Efficient Two Dimensional Range Minimum Data Structures, a data structure that answers queries in $O(1)$ time, after $O(NM)$ preprocessing.
Actually, we need to find the minimum of a rectangle without two of its corners, but this domain can be written as the union of three rectangles, so such a query can also be answered in constant time.
Using such a data structure, we obtain an algorithm running in linear time $O(NM)$.
In fact, it suffices to use a much simpler data structure supporting one-dimensional range minimum queries; see Fischer, Optimal Succinctness for Range Minimum Queries for appropriate references. Suppose that $N \leq M$. Using a range minimum query data structure on each row, we can answer a two-dimensional range minimum query in time $O(N)$. Since the algorithm above makes only $O(N+M) = O(M)$ such queries, the overall complexity is still $O(NM)$.
Not the answer you're looking for? Browse other questions tagged algorithms dynamic-programming lexical-analysis matrix or ask your own question.
How to optimize Dijkstra's algorithm for a grid graph? | CommonCrawl |
If we look at Euclidean and Manhattan distances, these are both just specific instances of $p = 2$ and $p=1$, respectively.
For $p < 1$ this distance measure is not actually a metric, but it may still be interesting sometimes. For this problem, write a program to compute the $p$-norm distance between pairs of points, for a given value of $p$.
The input file contains up to $1\, 000$ test cases, each of which contains five real numbers, $x_1~ y_1~ x_2~ y_2~ p$, each of which have at most $10$ digits past the decimal point. All coordinates are in the range $(0, 100]$ and $p$ is in the range $[0.1, 10]$. The last test case is followed by a line containing a single zero.
For each test case output the $p$-norm distance between the two points $(x_1,y_1)$ and $(x_2,y_2)$. Your answer may have absolute or relative error of at most $0.0001$. | CommonCrawl |
Abstract: We consider the real part $\Re(\zeta(s))$ of the Riemann zeta-function $\zeta(s)$ in the half-plane $\Re(s) \ge 1$. We show how to compute accurately the constant $\sigma_0 = 1.19\ldots$ which is defined to be the supremum of $\sigma$ such that $\Re(\zeta(\sigma+it))$ can be negative (or zero) for some real $t$. We also consider intervals where $\Re(\zeta(1+it)) \le 0$ and show that they are rare. The first occurs for $t$ approximately 682112.9, and has length about 0.05. We list the first fifty such intervals. | CommonCrawl |
Abstract: High dimensional data and systems with many degrees of freedom are often characterized by covariance matrices. In this paper, we consider the problem of simultaneously estimating the dimension of the principal (dominant) subspace of these covariance matrices and obtaining an approximation to the subspace. This problem arises in the popular principal component analysis (PCA), and in many applications of machine learning, data analysis, signal and image processing, and others. We first present a novel method for estimating the dimension of the principal subspace. We then show how this method can be coupled with a Krylov subspace method to simultaneously estimate the dimension and obtain an approximation to the subspace. The dimension estimation is achieved at no additional cost. The proposed method operates on a model selection framework, where the novel selection criterion is derived based on random matrix perturbation theory ideas. We present theoretical analyses which (a) show that the proposed method achieves strong consistency (i.e., yields optimal solution as the number of data-points $n\rightarrow \infty$), and (b) analyze conditions for exact dimension estimation in the finite $n$ case. Using recent results, we show that our algorithm also yields near optimal PCA. The proposed method avoids forming the sample covariance matrix (associated with the data) explicitly and computing the complete eigen-decomposition. Therefore, the method is inexpensive, which is particularly advantageous in modern data applications where the covariance matrices can be very large. Numerical experiments illustrate the performance of the proposed method in various applications. | CommonCrawl |
The retailer should charge 5 dollars for the candy.
Let s represent the selling price. We can use the following guideline to solve this problem: Selling price = Cost + Profit So: s = 2.5 + 50% $\times$ s s = 2.5 + 0.5 $\times$ s s = 2.5 + 0.5s 0.5s = 2.5 Divide both sides by 0.5: s = 5 The retailer should charge 5 dollars for the candy. | CommonCrawl |
Octonion algebras are non-associative algebras endowed with a non-degenerate multiplicative quadratic form, which over fields determines the algebra completely, but not over rings in general. Connecting to the talk given by P. Gille, and reporting on a joint work with him, I will show how to construct all octonion algebras having the same quadratic form. The key role is played by the triality phenomenon, as it relates certain $\mathbf G_2$-torsors to a classical construction of alternative algebras. | CommonCrawl |
$cl:2^X\to 2^X$ satisfying $A\subset cl(A)$ for all $A\subset X$. That's a good way to summarize Kuratowski's closure operator.
Basic geometry on a set $X$ is a dot product $\cdot:2^X\times 2^X\to 2^Y$. Its equivalent form is an orthogonality relation on subsets of $X$. The optimal case is if the orthogonality relation satisfies a variant of parallel-perpendicular decomposition from linear algebra.
Higson corona, Gromov boundary, \v Cech-Stone compactification, Samuel-Smirnov compactification, and Freudenthal compactification. | CommonCrawl |
How can I make large curly brackets spanning multiple lines?
Brackets and Parentheses Parentheses and brackets are very common in mathematical formulas. You can easily control the size and style of brackets in L a T e X ; this article explains how.
How do I add brackets to the beginning of every line?
Put multiple lines of text in one cell with pressing Alt + Enter keys You can put multiple lines in a cell with pressing Alt + Enter keys simultaneously while entering texts. Pressing the Alt + Enter keys simultaneously helps you separate texts with different lines in one cell.
Add parentheses to the following to make a true equation: $10-9\times8-7\times6-5\times4-3\times2-2\times1=1$ Stack Exchange Network Stack Exchange network consists of 174 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
A command list embedded between parentheses runs as a subshell. Variables in a subshell are not visible outside the block of code in the subshell. They are not accessible to the parent process, to the shell that launched the subshell.
If you don't want a multiline string but just have a long single line string, you can use parentheses, just make sure you don't include commas between the string segments, then it will be a tuple. | CommonCrawl |
In the above calculation, we calculated the determinant of the $3\times 3$ matrix by expanding along the third row. From the above equation, we can conclude that the eigenvalues of $A$ are $\lambda=3,1,-4$.
Previous story Find All Values of $a$ which Will Guarantee that $A$ Has Eigenvalues 0, 3, and -3. | CommonCrawl |
Logic is a part of the study of human reason, the ability we have to think abstractly, solve problems, explain the things that we know, and infer new knowledge on the basis of evidence. Traditionally, logic has focused on the last of these items, the ability to make inferences on the basis of evidence, by evaluating the deductive validity of arguments. This section explains briefly what this means.
M: An argument isn't just contradiction.
M: No it can't! An argument is a connected series of statements intended to establish a proposition.
M: Yes it is! 'tisn't just contradiction.
In logic, we use the word 'argument' to refer to the attempt to show that certain evidence supports a conclusion. This is very different from the sort of argument you might have with your family, which could involve screaming and throwing things. We are going to use the word 'argument' a lot in this book, so you need to get used to thinking of it as a name for a rational process, and not a word that describes what happens when people contradicts or disagree with each other. .
An argument in this technical is a set of statements intended to provide someone a reason to believe one of the statements in that set, which we call a conclusion. Suppose you are wondering if you friend Bob plays the guitar. You can't quite remember exactly what he plays, but you do recall that he uses a bow. Since you know that guitars are plucked and not bowed, you conclude that Bob indeed does not play the guitar.
We can reconstruct this line of thought as an argument, like the one to the right. We call the two sentences above line premises. The word 'therefore' signifies that the sentence below the line is the $conclusion$ of the argument. If you believe the premises, then the argument provides you with a reason to believe the conclusion. You might use reasoning like this purely in your own head, without talking with anyone else. Sometimes you might work things verbally. This is not a problem, because the business of logic is not to describe exact what's going on in your mind but to systemize the rational structure of thoughts.
1: Bob plays a bowed instrument.
The important thing to see is that the definition above tries to get at what $would$ happen if the premises were true. It does not assert that the premises actually $are$ true. (It also does not assert that they are false.)This is why a valid argument is sometimes defined as one where the conclusion is true in every imaginable scenario in which the premises are true. This sounds pretty imprecise at the moment, but we will sharpen our understanding of validity as we go on.
Evaluating arguments based on their validity is called deductive reasoning, in contradistinction to inductive reasoning, where the value of an inference is based on its probability. In this class we concern ourselves exclusively with deductive reasoning. The techniques you would learn in a probability or statistics class are examples of inductive reasoning.
One way in which we can think about validity is by representing an argument like the one above in a graphical form that accentuates its logical structure. First, consider a circle that contains all plucked instruments and another one that contains all bowed instruments.
states that whatever instrument Bob plays belongs to the bowed circle. This can be represent by a smaller circle situated within the bowed circle. Based on this premise alone, do we have good enough reason to think that Bob does not play the guitar?
Since that premise alone does not give us enough information to where guitars would be in this diagram. Premise 1 only implies that Bob's instrument is in the bowed circle - but the overlapping area is still part of that circle! What if guitars can be both plucked and bowed? Then it would belong in the middle. Premise 1 $by$ $itself$ does not discriminate against that.
guitars are not played with bows. This means that it is impossible for the guitar circle to be inside of the overlapping areas.
Diagrams like these are immensely useful at the beginning of the study of logic. We will use them for quite a bit in the first module. There are some nuances to them, however, so we will discuss them more thoroughly. Let us finish this section with a short exercise.
Instruction: Make your best estimate as to whether the arguments presented below are valid.
Note: not currently logged in!
For this reading exercise, try to find out if the argument given is valid, in the technical sense defined above: if the premises is true, must the conclusion be true as well?
It might be difficult to think about these arguments just in your head. Just try your best - for this exercise you are not graded on correctness. You will get credit as long as you complete the set of 10 questions.
After you are done, you can look at the answers. These problems are randomly generated, so you can redo it for practice.
Consider this argument: 'No Huskies are Bulldogs. All dogs are Bulldogs. Conclusion: no Huskies are dogs.' To begin, it's obvious that something is off with the argument: one of the premises 'all dogs are Bulldogs' is clearly false. Does that mean the argument is invalid?
Not necessarily! Remember an argument is valid $if$ the premises were true, the conclusion would be true as well. So there is a hypothetical nature to validity - we are to ask if the premises are true: we are asking what would happen if they were. So this argument is indeed valid - IF all dogs are Bulldogs, then there is no way for Huskies to be dogs.
Because of this, there is another concept for valid arguments that contain true premises called soundness. An argument is sound when it is valid AND has only true premises.
To be sure to keep this in mind when doing this reading exercise. You only have to do it once to get credit for it. You are not graded by your performance, but you are encouraged to do well on them. Once you are finished, correct answers will be shown. It would be a good idea to check and see what you have gotten wrong.
I hope by now you have gotten a sense of what validity means and how it could be something that is difficult to think about intuitively. Most of what we do here is to learn formal tools that allow us to evaluate the validity of complex arguments more efficiently and reliably. Venn diagrams, the topic of the next section, is one of them.
In 1880 English logician John Venn published two essays on the use of diagrams with circles to represent categorical propositions, like the ones in our Bob example. Venn noted that the best use of such diagrams so far had come from the brilliant Swiss mathematician Leonhard Euler, but they still had many problems, which Venn felt could be solved by bringing in some ideas about logic from his fellow English logician George Boole. Although Venn only claimed to be building on the long logical tradition he traced, since his time these kinds of circle diagrams have been known as Venn diagrams. While they might not be as elegant and powerful as some of the tools we will learn later on, they are still useful in evaluating arguments.
Consider the diagram below that represents the claim that all guitars are plucked instruments. Outside of college logic classes, you may have seen people use a diagram like this to represent a situation where one group is a subclass of another. You may have even seen people call concentric circles like this a Venn diagram. But Venn did not think we should put one circle entirely inside the other if we just want to represent 'All X is Y.' Thus, technically speaking what we have here is an Euler diagrams, a precursor of the Venn diagram.
What's wrong with Euler diagrams?
So it should leave it open whether the guitar circle is smaller than or the same size as the plucked circle. The problem however is that Euler diagrams cannot express this relation clearly - either the two circles are of the size, or one is smaller. There is no way to express the ambiguity needed here. Furthermore, it is confusing to put one of the circles directly on top of the other, as shown by the diagram.
to represent just the content of a single proposition, we should always begin by drawing partially overlapping circles. The advantage of this is that we always have spaces available to represent the four possible ways the terms can combine: Area 1 represents things that are plucked instruments but not guitars; area 2, things that are plucked instruments and guitars; area 3, things that are just guitars; and area 4 represents things that are neither plucked instruments nor guitars.
We can then mark up these areas to indicate whether something is there or could be there. We shade a region of the diagram to represent the claim that nothing can exist in that region. For instance, if we say 'All guitars are plucked instruments,' we are asserting that nothing can exist that is in the guitar circle unless it is also in the plucked instruments circle. So we shade out the part of the guitar circle that doesn't overlap with plucked instruments.
Instruction: Based on the given Venn Diagram, answer the corresponding question.
The premise 'all guitars are plucked instruments' is an example of what logicians call a categorical statement. For most of the history of logic in the West, the focus has been on arguments that rely extensively on those statements called categorical syllogism . Aristotle began the study of this kind of argument in his book the Prior Analytics (c.350 BCE).
A categorical syllogism is a two-premise argument composed of categorical statements. There are actually all kinds of two-premise arguments using categorical statements, but Aristotle only looked at arguments where each statement is in one of the 'moods' of categorical statement. Each mood is a combination of two quantities and two qualities. A quantity of a categorical statement can be either universal - applying to everything in a category, particular - at least one thing in a category. Each categorical statement is also said to have either an affirmative or a negative quality. 'All guitars are plucked' is an instance of a universal affirmation - it makes a positive claim that everything that in the category of guitars. 'No guitar is bowed' would be a instance of universal negation. Some guitars are bowed would be an instance of a 'particular affirmation', and so on.
If a region of a Venn diagram is blank - if it is neither shaded nor has an x in it - then it could go either way. Maybe such things exist, maybe they do not. Notice that when we draw diagrams for the two universal forms, we do not draw any x's. For these forms we are only ruling out possibilities, not asserting that things actually exist. This is part of what Venn learned from Boole. The proposition, 'All guitars are plucked instruments,' denies the existence of any guitar that is not a plucked, but it does not assert the existence of some guitar that is a plucked. That probably reads like gibberish - guitars obviously do exist. So what's the deal?
The reason for this is to accommodate categorical statements about things that don't exist yet makes perfect sense, for instance 'All unicorns have one horn.' This seems like a true statement, but unicorns don't exist. Perhaps what we mean by 'All unicorns have one horn' is that if a unicorn existed, then it would have one horn. But if we interpret the statement about unicorns that way, shouldn't we also interpret the statement about dogs that way? Really all we mean when we say 'All dogs are mammals' is that if there were dogs, then they would be mammals. It takes an extra assertion to point out that dogs do, in fact, exist.
The issue we are discussing here is called existential import. A sentence is said to have existential import if it asserts the existence of the things it is talking about. Until Boole, universal-affirmative statements were often interpreted as having existential import. You might find that more intuitive, but if you interpret all universal-affirmative statements with existential import, they are always false when you are talking about things that don't exist. So, 'All unicorns have one horn' is false in the traditional interpretation. On the other hand, in the modern interpretation all statements about things that don't exist are true. 'All unicorns have one horn' simply asserts that there are no multi-horned unicorns, and this is true because there are no unicorns at all. We call this vacuous truth. Something is vacuously true if it is true simply because it is about things that don't exist. Note that all statements about nonexistent things become vacuously true if you assume they have no existential import, even a statement like 'All unicorns have more than one horn.' A statement like this simply rules out the existence of unicorns with one horn or fewer, and these don't exist because unicorns don't exist. This is a complicated issue that will come up again starting in later sections when we consider conditional statements in this module, and predicate logic later.
P1. All mammals are vertebrates.
P2. All dogs are mammals.
C. All dogs are vertebrates.
Notice how the statements in this argument overlap each other. Each statement shares a term with the other two. Premise 2 shares a term with the conclusion and another with Premise 1. Thus there are only three terms spread across the three statements. Each of the three statements can take one of four categorical mood. This gives us $4 \times 4 \times 4,$ or 64 possibilities. In addition to varying the kind of statements we use in an Aristotelian syllogism, we can also vary the placement of the terms involved. The combination of 64 moods and 4 figures gives us a total of 256 possible Aristotelian syllogisms. Most of these are valid, but a good chunk of them are. We won't go through every single syllogism like a good medieval logician, but we should be able to analyze them using Venn diagram.
Since each syllogism involves three terms, we need 3 overlapping circles.
All mammals are vertebrates, so we grey out the the area of the mammal circle that does not overlap with the vertebrate circle.
All dogs are mammals, so we grey out the the area of the dog circle that does not overlap with the mammal circle. Note that part of the circle was greyed already due to premise 1.
Does the conclusion 'all dogs are vertebrates' follow?
Upon examining this Venn diagram, we can see that if we greyed out areas in accordance to the premises, the only white area left in the dog circle also overlaps with the vertebrate circle. This represents the idea of an valid argument as one with premises that, if true, would make the conclusion true as well.
Instruction: Match the following definitions by dragging the definition to the box containing the concept.Note: not currently logged in!
Finish this section by completing the concepts review quiz. For this quiz, you have to answer all questions correctly to get credit. However you may try as many times as you like.
After familiarizing yourself with the ideas in the section, you should do the logical exercise ('logicise') 'Venn Diagrams and Syllogistic Validity,' where you will be asked to build Venn diagrams to determine the validity of categorical arguments.
We actually will not go any further into categorical arguments. While categorical syllogism is taught in many contexts and important for historical reasons, it is very limited compared to the formal system we will learn in this class. Nevertheless, I hope it gave a you taste of the sort of stuff we will be doing for the rest of the course. But the ideas involved in categorical logic, such as existence and categories are still very important, especially in our later study of predicate logic in module 2. Before we go there, however, we will learn about the foundation of modern logic - the formal language of sentence logic. | CommonCrawl |
Abstract: We study the asymptotic consistency properties of $\alpha$-Rényi approximate posteriors, a class of variational Bayesian methods that approximate an intractable Bayesian posterior with a member of a tractable family of distributions, the member chosen to minimize the $\alpha$-Rényi divergence from the true posterior. Unique to our work is that we consider settings with $\alpha > 1$, resulting in approximations that upperbound the log-likelihood, and consequently have wider spread than traditional variational approaches that minimize the Kullback-Liebler (KL) divergence from the posterior. Our primary result identifies sufficient conditions under which consistency holds, centering around the existence of a `good' sequence of distributions in the approximating family that possesses, among other properties, the right rate of convergence to a limit distribution. We also further characterize the good sequence by demonstrating that a sequence of distributions that converges too quickly cannot be a good sequence. We also illustrate the existence of good sequence with a number of examples. As an auxiliary result of our main theorems, we also recover the consistency of the idealized expectation propagation (EP) approximate posterior that minimizes the KL divergence from the posterior. Our results complement a growing body of work focused on the frequentist properties of variational Bayesian methods. | CommonCrawl |
This is an initiation project to introduce RAMP and get you to know how it works.
The goal is to develop prediction models able to identify which news is fake.
You goal is to classify each statement (+ metadata) into one of the categories.
The original training data frame has 13000+ instances. In the starting kit, we give you a subset of 7569 instances for training and 2891 instances for tesing.
Most columns are categorical, some have high cardinalities.
If you want to use the journalist and the editor as input, you will need to split the lists since somtimes there are more than one of them on an instance.
"Angie Drobnic Holan, Louis Jacobson, Ciara O'Rourke"
"Becky Bowers, Louis Jacobson, Erin O'Neill, Bill Wichert"
'Chris Joyner, Karishma Mehrotra' 'Christian Gaston' "Ciara O'Rourke"
'Erin McNeill' 'Erin Mershon' "Erin O'Neill" "Erin O'Neill, Bill Wichert"
"Erin O'Neill, Caryn Shinske, Bill Wichert"
'Louis Jacobson, Carol Rosenberg' "Louis Jacobson, Ciara O'Rourke"
"Meghan Ashford-Grooms, Ciara O'Rourke, W. Gardner Selby"
'Robert Farley, Catharine Richert' "Robert Farley, Ciara O'Rourke"
There are 2000+ different sources.
the class FeatureExtractor, which will be used to extract features for classification from the dataset and produce a numpy array of size (number of samples $\times$ number of features).
The feature extractor implements a transform member function. It is saved in the file submissions/starting_kit/feature_extractor.py. It receives the pandas dataframe X_df defined at the beginning of the notebook. It should produce a numpy array representing the extracted features, which will then be used for the classification.
Note that the following code cells are not executed in the notebook. The notebook saves their contents in the file specified in the first line of the cell, so you can edit your submission before running the local test below and submitting it at the RAMP site.
"""Convert a collection of raw documents to a matrix of TF-IDF features. """
"""Learn a vocabulary dictionary of all tokens in the raw documents.
An iterable which yields either str, unicode or file objects.
First, we preprocess the text. Preprocessing text is called tokenization or text normalization.
The first step or preprocessing. Split sentences in words.
The most frequent words often do not carry much meaning. Examples: the, a, of, for, in, .... This stopword list can be found in NLTK library stopwords.words('english'). Throw away unwanted stuf as in ["`", "", "..."] or numbers.
This is optional. English words like look can be inflected with a morphological suffix to produce looks, looking, looked. They share the same stem look. Often (but not always) it is beneficial to map all inflected forms into the stem. The most commonly used stemmer is the Porter Stemmer. The name comes from its developer, Martin Porter. SnowballStemmer('english') from NLTK is used. This stemmer is called Snowball, because Porter created a programming language with this name for creating new stemming algorithms.
Before going through the code, we first need to understand how tf-idf works. A Term Frequency is a count of how many times a word occurs in a given document (synonymous with bag of words). The Inverse Document Frequency is the the number of times a word occurs in a corpus of documents. tf-idf is used to weight words according to how important they are. Words that are used frequently in many documents will have a lower weighting while infrequent ones will have a higher weighting.
class FeatureExtractor(TfidfVectorizer) inherits a TfidfVectorizer which is a CountVectorizer followed by TfidfTransformer.
CountVectorizer converts a collection of text documents to a matrix of token (word) counts. This implementation produces a sparse representation of the counts to be passed to the TfidfTransformer. The TfidfTransformer transforms a count matrix to a normalized tf or tf-idf representation.
A TfidfVectorizer does these two steps.
The feature extractor overrides fit by provinding the TfidfVectorizer with a new preprocessing step that has been presented before.
The classifier follows a classical scikit-learn classifier template. It should be saved in the file submissions/starting_kit/classifier.py. In its simplest form it takes a scikit-learn pipeline, assigns it to self.clf in __init__, then calls its fit and predict_proba functions in the corresponding member funtions.
If it runs and print training and test errors on each fold, then you can submit the code.
Once you found a good feature extractor and classifier, you can submit them to ramp.studio. First, if it is your first time using RAMP, sign up, otherwise log in. Then find an open event on the particular problem, for example, the event fake_news (Saclay Datacamp, DataFest Tbilisi) for this RAMP. Sign up for the event. Both signups are controled by RAMP administrators, so there can be a delay between asking for signup and being able to submit.
Once your signup request is accepted, you can go to your sandbox (Saclay Datacamp, DataFest Tbilisi) and copy-paste (or upload) feature_extractor.py and classifier.py from submissions/starting_kit. Save it, rename it, then submit it. The submission is trained and tested on our backend in the same way as ramp_test_submission does it locally. While your submission is waiting in the queue and being trained, you can find it in the "New submissions (pending training)" table in my submissions (Saclay Datacamp, DataFest Tbilisi). Once it is trained, you get a mail, and your submission shows up on the public leaderboard (Saclay Datacamp, DataFest Tbilisi). If there is an error (despite having tested your submission locally with ramp_test_submission), it will show up in the "Failed submissions" table in my submissions (Saclay Datacamp, DataFest Tbilisi). You can click on the error to see part of the trace.
The official score in this RAMP (the first score column after "historical contributivity" on the leader board (Saclay Datacamp, DataFest Tbilisi) is smoothed accuracy, so the line that is relevant in the output of ramp_test_submission is valid sacc = 0.361 ± 0.05. When the score is good enough, you can submit it at the RAMP. | CommonCrawl |
What is the difference between a qubit and classical bit?
As I understand it, the main difference between quantum and non-quantum computers is that quantum computers use qubits while non-quantum computers use (classical) bits.
What is the difference between qubits and classical bits?
A bit is a binary unit of information used in classical computation. It can take two possible values, typically taken to be $0$ or $1$. Bits can be implemented with devices or physical systems that can be in two possible states.
To compare and contrast bits with qubits, let's introduce a vector notation for bits as follows: a bit is represented by a column vector of two elements $(\alpha,\beta)^T$, where $\alpha$ stands for $0$ and $\beta$ for $1$. Now the bit $0$ is represented by the vector $(1,0)^T$ and the bit $1$ by $(0,1)^T$. Just like before, there are only two possible values.
While this kind of representation is redundant for classical bits, it is now easy to introduce qubits: a qubit is simply any $(\alpha,\beta)^T$ where the complex number elements satisfy the normalization condition $|\alpha|^2+|\beta|^2=1$. The normalization condition is necessary to interpret $|\alpha|^2$ and $|\beta|^2$ as probabilities for measurement outcomes, as will be seen. Some call qubit the unit of quantum information. Qubits can be implemented as the (pure) states of quantum devices or quantum systems that can be in two possible states, that will form the so called computational basis, and additionally in a coherent superposition of these. Here the quantumness is necessary to have qubits other than the classical $(1,0)^T$ and $(0,1)^T$.
The usual operations that are carried out on qubits during a quantum computation are quantum gates and measurements. A (single qubit) quantum gate takes as input a qubit and gives as output a qubit that is a linear transformation of the input qubit. When using the above vector notation for qubits, gates should then be represented by matrices that preserve the normalization condition; such matrices are called unitary matrices. Classical gates may be represented by matrices that keep bits as bits, but notice that matrices representing quantum gates do not in general satisfy this requirement.
A measurement on a bit is understood to be a classical one. By this I mean that an a priori unknown value of bit can in principle be correctly found out with certainty. This is not the case for qubits: measuring a generic qubit $(\alpha,\beta)^T$ in the computational basis $[ (1,0)^T,(0,1)^T]$ will result in $(1,0)^T$ with probability $|\alpha|^2$ and in $(0,1)^T$ with probability $|\beta|^2$. In other words, while qubits can be in states other than computational basis states before measurement, measuring can still have only two possible outcomes.
There is not much one can do with a single bit or qubit. The full computational power of either comes from using many, which leads to the final difference between them that will be covered here: multiple qubits can be entangled. Informally speaking, entanglement is a form of correlation much stronger than classical systems can have. Together, superposition and entanglement allow one to design algorithms realized with qubits that cannot be done with bits. Of greatest interest are algorithms that allow the completion of a task with reduced computational complexity when compared to best known classical algorithms.
Before concluding, it should be mentioned that a qubit can be simulated with bits (and vice versa), but the number of bits required grows rapidly with the number of qubits. Consequently, without reliable quantum computers quantum algorithms are of theoretical interest only.
Not the answer you're looking for? Browse other questions tagged physical-qubit or ask your own question.
Can a Turing machine simulate a quantum computer?
Can a quantum computer simulate a normal computer?
How does using a superposition of 0 and 1 improve the processing capabilities of a quantum computer compared to classical computers?
What is the physical representation of a qubit?
What is the difference between transmon and Xmon qubits?
What is the relation between single photon qubits and squeezed light qubits?
What's the difference between a set of qubits and a capacitor with a subdivided plate?
How is a single qubit fundamentally different from a classical coin spinning in the air?
Difference between coherence transfer, polarization transfer and population transfer?
Non-layperson explanation of why a qubit is more useful than a bit? | CommonCrawl |
We introduce a new online deadlock-avoidance policy, Transitive Joins (TJ), that targets programs with dynamic task parallelism and arbitrary join operations. In this model, a computation task can asynchronously spawn new tasks and selectively join (block) on any task for which it has a handle. We prove that TJ soundly guarantees the absence of deadlock cycles among the blocking join operations. We present an algorithm for dynamically verifying TJ and show that TJ results in fewer false positives than the state-of-the-art policy, Known Joins (KJ). We evaluate an implementation of our verifier in comparison to prior work. The evaluation results show that instrumenting a program with a TJ verifier incurs geometric mean overheads of only 1.06$\times$ in execution time and 1.09$\times$ in memory usage, which is better overall than existing KJ verifiers. TJ is a practical online deadlock-avoidance policy that is applicable to a wide range of parallel programming models. | CommonCrawl |
Aims. Photodissociation by UV light is an important destruction mechanism for carbon monoxide (CO) in many astrophysical environments, ranging from interstellar clouds to protoplanetary disks. The aim of this work is to gain a better understanding of the depth dependence and isotope-selective nature of this process. Methods. We present a photodissociation model based on recent spectroscopic data from the literature, which allows us to compute depth-dependent and isotope-selective photodissociation rates at higher accuracy than in previous work. The model includes self-shielding, mutual shielding and shielding by atomic and molecular hydrogen, and it is the first such model to include the rare isotopologues C17O and 13C17O. We couple it to a simple chemical network to analyse CO abundances in diffuse and translucent clouds, photon-dominated regions, and circumstellar disks. Results. The photodissociation rate in the unattenuated interstellar radiation field is 2.6 $\times$ 10-10 s-1, 30% higher than currently adopted values. Increasing the excitation temperature or the Doppler width can reduce the photodissociation rates and the isotopic selectivity by as much as a factor of three for temperatures above 100 K. The model reproduces column densities observed towards diffuse clouds and PDRs, and it offers an explanation for both the enhanced and the reduced N(12CO)/N(13CO) ratios seen in diffuse clouds. The photodissociation of C17O and 13C17O shows almost exactly the same depth dependence as that of C18O and 13C18O, respectively, so 17O and 18O are equally fractionated with respect to 16O. This supports the recent hypothesis that CO photodissociation in the solar nebula is responsible for the anomalous 17O and 18O abundances in meteorites. Grain growth in circumstellar disks can enhance the N(12CO)/N(C17O) and N(12CO)/N(C18O) ratios by a factor of ten relative to the initial isotopic abundances. | CommonCrawl |
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A couple of days ago, I received the following email from Home office(UK) for my spouse visa. I have deposited my IHS fee and it's been a week since I submitted my passport. Is there any chance where they can still refuse my visa? Additionally, how long would it take for them to return my passport?
How do I view the files on my tablet from my phone? Do I have to connect wirelessly? I am working on a translation and would like to see the original on one tablet and the file I'm working on on the smartphone or other tablet.
Do I need an app or is this service part of the OS? I would like to just use Bluetooth.
I was able to get my custom font to my theme. I made a style to use it, but the text-transform and letter-spacing attributes are being inherited from another style sheet? What do I need to do to make sure that my ss_logo style always wins?
Please forgive my ignorance. I haven't done any web-type development for over 10 years.
A common algorithm used in N-body gravitational calculations with large numbers of bodies (stars in a galaxy, galaxies in the universe, etc.) is the Barnes-Hut algorithm, which assembles particles into an octree for approximate bulk calculations of gravitational forces between distant areas. The complexity of this algorithm is O(N log N), compared to the O(N^2) of a more realistic direct calculation between all pairs of points.
I'm trying to fully understand where the N log N comes from. I understand that, once the octree is assembled, the calculation of forces is N log N because you have to go through each particle (N), and each particle "sees" approximately log N other particles because of the octree reducing the number of calculations needing to be performed for distant particles.
What I'm still trying to understand is what the complexity is for assembling the octree in the first place. Is it also N log N? N because you need to do it for each particle, and log N because that's approximately how deep you'll have to go (with some factor in front) to reach an isolated leaf in the tree?
I did repair my early 2009 mac pro 4.1; and after installing a gtx970 I did realize that I may use it also for windows gaming.
Although my main drive is just a boot for OSX, which is a 90 GB SSD drive; the rest of my home directory and application support is on a SATA mechanical drive. Is possible to install Windows 10 via bootcamp, on a drive that is not the main drive? I am running 10.11 (el capitan), which is the highest OS that my machine seems to be able to run.
How to create a "read more" button using AJAX in a module?
I'm working on a really simple module in Drupal 8 to get the hang of things. What it does (or rather, what the page I'm asking about in this question does) is grab some data of a bunch of simple article-style entries, and display them.
What I would like to be able to do is to have it only display the first few (say, 5), and then have a "Read more" button, which uses AJAX to display the next 5, lengthening the page so it now shows 10.
Here is the controller as I have it at the moment. The routing.yml file points to MyController::build. For the sake of simplicity, I've left out exactly how it gets things from the database, and some simple processing (in particular, image gets converted from a managed_file in the database into a URL to the image).
What I want is a way to change this set up so that at first the controller only sends a few of the entries, but when the button is pushed, it sends the next few, and the template displays the new ones as well as the old.
to the bottom of the .html.twig file.
I created a file identical to his ReadMessageCommand.php, except named mine MyMessageCommand.php (and renamed the class inside it accordingly). The function render has myMessage in place of readMessage.
I added a method called myMessageCallback into MyController, which works the same as in that guide, but with my_module_load_message and MyMessageCommand in the appropriate places.
To readmore.js, I added the function from the above guide, but with readMessage replaced by myMessage. I also added the following, just so I can know more easily if the function is at least being called.
And I added the core/drupal.ajax dependency to the readmore library.
My question is, how do I actually get this function to be called. As far as I can tell, the guide that I've used has no indication of it. How to I get a message to be displayed in the div at the bottom? Once I've figured that out, I feel extending it so that the button acts as a "read more" shouldn't be too much trouble.
Suppose I have a 5×5 grid of squares. I would like to fill in 15 checkmarks in the squares such that (1) each of the 25 square cells contains at most one checkmark, (2) each row has exactly 3 checkmarks, and (3) each column has exactly 3 checkmarks. How many ways are there to fill in these 15 checkmarks?
More generally, suppose I have an $ n \times n $ square grid, and I would like to fill in $ mn$ checkmarks such that (1) each of the $ n^2$ square cells contains at most one checkmark, (2) each row has exactly m checkmarks, and (3) each column has exactly m checkmarks. How many ways are there to do so? If $ m=1,$ I think the answer is $ n!$ . But I am not sure about the general case.
Also, if I have an additional restriction that no checkmarks on the diagonal, i.e., no checkmark in the (1,1), (2,2),… (n,n) cells. How many ways are there?
Thanks very much! Wish all very happy new year!
I am DMing Lost Mines of Phandelver, and it's looking like the party wants to keep the cave as a base to replace the BBEG and turn evil.
Main question is: if I decide to let them fix up the forge of spells after a few quests for parts, what level / how challenging should this be?
I love the rule of unintended consequences, but am not sure of any down sides. | CommonCrawl |
Abstract: We study the fluctuations of the spin per site around the thermodynamic magnetization in the mean-field Blume-Capel model. Our main theorem generalizes the main result in a previous paper (Ellis, Machta, and Otto) in which the first rigorous confirmation of the statistical mechanical theory of finite-size scaling for a mean-field model is given. In that paper our goal is to determine whether the thermodynamic magnetization is a physically relevant estimator of the finite-size magnetization. This is done by comparing the asymptotic behaviors of these two quantities along parameter sequences converging to either a second-order point or the tricritical point in the mean-field Blume-Capel model. The main result is that the thermodynamic magnetization and the finite-size magnetization are asymptotic when the parameter $\alpha$ governing the speed at which the sequence approaches criticality is below a certain threshold $\alpha_0$. Our main theorem in the present paper on the fluctuations of the spin per site around the thermodynamic magnetization is based on a new conditional limit theorem for the spin, which is closely related to a new conditional central limit theorem for the spin. | CommonCrawl |
Now we don't need to consider $1\times 1$ any longer as we have found the smallest rectangle tilable with copies of U plus copies of $1\times 1$.
There are at least 6 more solutions. I tagged it 'computer-puzzle' but you can certainly work some of these out by hand. The larger ones might be a bit challenging. | CommonCrawl |
Say I fit a multiple regression of p explanatory variables. The t-test will allow me to check if any single one of those is significant ($H_0: \beta_i = 0$). I can do a partial F-test to check if some subset of them is significant ($H_0: \beta_i=\beta_j=...=\beta_k=0$).
What I often see though is that someone gets 5 p-values from 5 t-tests (assuming they had 5 covariates) and only keeps the ones with a p-value < 0.05. That seems a bit incorrect as there really should be a multiple comparison check no? Is it really fair to say something like $\beta_1$ and $\beta_2$ are significant but $\beta_3$, $\beta_4$ and $\beta_5$ are not?
On a related note, say I run 2 regressions on 2 separate models (different outcome). Does there need to be a multiple comparison check for significant parameters between the two outcomes?
Edit: To differentiate from the similar question, is there any other interpretation to the p-values besides: "B_i is (in)significant, when adjusting for all the other covariates"? It doesn't seem that this interpretation allows me to look at every B_i and drop those less than 0.5 (which is similar to the other post).
It seems to me that a sure fire way to test whether B_i and Y have an a relationship would be to get a correlation coefficient p-value for each covariate and then do a multcomp (although that would definitely lose signal).
Finally, say I computed the correlation between B1/Y1, B2/Y1 and B3/Y1 (thus three p-values). Unrelatedly, I also did a correlation between T1/Y2, T2/Y2, T3/Y2. I'm assuming the correct Bonferroni adjustment would be 6 for all 6 tests together (rather than 3 for the first group and 3 for the second group - and thus getting 2 "semi"-adjusted p-values).
You're right. The problem of multiple comparisons exists everywhere, but, because of the way it's typically taught, people only think it pertains to comparing many groups against each other via a whole bunch of $t$-tests. In reality, there are many examples where the problem of multiple comparisons exists, but where it doesn't look like lots of pairwise comparisons; for example, if you have a lot of continuous variables and you wonder if any are correlated, you will have a multiple comparisons problem (see here: Look and you shall find a correlation).
Another example is the one you raise. If you were to run a multiple regression with 20 variables, and you used $\alpha=.05$ as your threshold, you would expect one of your variables to be 'significant' by chance alone, even if all nulls were true. The problem of multiple comparisons simply comes from the mathematics of running lots of analyses. If all null hypotheses were true and the variables were perfectly uncorrelated, the probability of not falsely rejecting any true null would be $1-(1-\alpha)^p$ (e.g., with $p=5$, this is $.23$).
The first strategy to mitigate against this is to conduct a simultaneous test of your model. If you are fitting an OLS regression, most software will give you a global $F$-test as a default part of your output. If you are running a generalized linear model, most software will give you an analogous global likelihood ratio test. This test will give you some protection against type I error inflation due to the problem of multiple comparisons (cf., my answer here: Significance of coefficients in linear regression: significant t-test vs non-significant F-statistic). A similar case is when you have a categorical variable that is represented with several dummy codes; you wouldn't want to interpret those $t$-tests, but would drop all dummy codes and perform a nested model test instead.
Regarding the issue of using $p$-values to do model selection, I think this is a really bad idea. I would not move from a model with 5 variables to one with only 2 because the others were 'non-significant'. When people do this, they bias their model. It may help you to read my answer here: algorithms for automatic model selection to understand this better.
Regarding your update, I would not suggest you assess univariate correlations first so as to decide which variables to use in the final multiple regression model. Doing this will lead to problems with endogeneity unless the variables are perfectly uncorrelated with each other. I discussed this issue in my answer here: Estimating $b_1x_1+b_2x_2$ instead of $b_1x_1+b_2x_2+b_3x_3$.
With regard to the question of how to handle analyses with different dependent variables, whether you'd want to use some sort of adjustment is based on how you see the analyses relative to each other. The traditional idea is to determine whether they are meaningfully considered to be a 'family'. This is discussed here: What might be a clear, practical definition for a "family of hypotheses"? You might also want to read this thread: Methods to predict multiple dependent variables.
On a practical level, I think one needs to also consider if the Betas reflect the levels of a categorical variables (i.e. dummies). In these circumstances it's reasonable to be interested in knowing whether a given Beta is different compared to a (meaningful) referent Beta. But before even doing pairwise comparisons, one would need to know whether overall the levels of the categorical variable are important (using a joint F test or a likelihood ratio test). Doing this has the advantage of using less d.f.
Not the answer you're looking for? Browse other questions tagged multiple-regression multiple-comparisons or ask your own question.
What might be a clear, practical definition for a "family of hypotheses" (with respect to familywise error rate)?
Are multiple comparisons corrections necessary for informal/visual "multiple comparisons"?
Multiple comparisons for correlation matrix?
How should I correct for multiple comparisons?
Why conduct large scale multiple comparisons rather than multiple regression?
Which of these scenarios are multiple or repeated comparisons? | CommonCrawl |
There's a nice paper by Menon and Elkan about dyadic prediction with latent features. It cares about the right things: ease of implementation, ability to incorporate multiple sources of information, and scalability. The model can be thought of as a mash-up of multinomial logit and matrix factorization.
\] where $\alpha$ is a vector of $k$ latent factors associated with $r$ and $\beta$ is a vector of $k$ latent factors associated with $c$. $r$ and $c$ are identifiers here, e.g., user ids and movie ids, user ids and advertisement ids, etc.
and then choose --quadratic ab and --loss logistic. Unfortunately this does not do the right thing. First, it creates some extra features (it is a outer product, not an inner product). Second, these extra features have their own independent weights, whereas in the model the weight is the product of the individual weights. A possible solution is to add a --dotproduct option to vowpal which would take two namespaces and emulate the features corresponding to their inner product (in this case the order of the features in the input would matter).
If you've followed this far, you can probably see how additional features $s (x)$ associated with the dyad can be added in another namespace to augment the model with side information.
Similarly it is easy to incorporate side information associated with each component, which would not be placed inside the alpha and beta namespaces to avoid getting picked up by the --dotproduct (in fact, for typical side information associated with the components, --quadratic on the component side information would be reasonable). Note the authors report better results learning the latent model first, fixing the latent weights, and then learning the weights associated with the side information.
For multi-valued data the authors use multinomial logit but I suspect a scoring filter tree with a logistic base learner could get the job done.
Finally, the authors suggest that regularization is necessary for good results. Possibly I can get away with using the ``only do one pass through the training data'' style of regularization. | CommonCrawl |
What's the point of Pauli's Exclusion Principle if time and space are continuous?
What does the Pauli Exclusion Principle mean if time and space are continuous?
If time and space are continuous then identical quantum states are impossible to begin with. in the question.
This assertion is just plainly false. A quantum state is not given by a location in time and space. The often used kets $\lvert x\rangle$ that are "position eigenstates" are not actually admissible quantum states since they are not normalized - they do not belong to the Hilbert space of states. Essentially by assumption, the space of states is separable, i.e. spanned by a countably infinite orthonormal basis.
Real particles are never completely localised in space (well except in the limit case of a completely undefined momentum), due to the uncertainty principle. Rather, they are necessarily in a superposition of a continuum of position and momentum eigenstates (a wave packet).
Pauli's Exclusion Principle asserts that they cannot be in the same exact quantum state, but a direct consequence of this is that they tend to also not be in similar states. This amounts to an effective repulsive effect between particles.
You can see this by remembering that to get a physical two-fermion wavefunction you have to antisymmetrize it. This means that if the two single wavefunctions are similar in a region, the total two-fermion wavefunction will have nearly zero probability amplitude in that region, thus resulting in an effective repulsive effect.
As you can clearly see for this picture, for $x_1=x_2$ the probability vanishes, as an immediate consequence of Pauli's exclusion principle: you cannot find the two identical fermions in the same position state. But you also see that the more $x_1$ is close to $x_2$ the smaller is the probability, as it must be due to the wavefunction being continuous.
Addendum: Can the effect of Pauli's exclusion principle be thought of as a force in the conventional $F=ma$ sense?
The QM version of what is meant by force in the classical setting is an interaction mediated by some potential, like the electromagnetic interaction between electrons. This is in practice an additional term in the Hamiltonian of the system, which says that certain states (say, same charges very close together) correspond to high-energy states and are therefore harder to reach, and vice versa for low-energy states.
Pauli's exclusion principle is conceptually entirely different: it is not due to an increase of energy associated with identical fermions being close together, and there is no term in the Hamiltonian that mediates such "interaction" (important caveat here: this "exchange forces" can be approximated to a certain degree as "regular" forces).
Rather, it comes from the inherently different statistics of many-fermion states: it is not that identical fermions cannot be in the same state/position because there is a repulsive force preventing it, but that there is no physical (many-body) state associated with them being in the same state/position. There simply isn't: it's not something compatible with the physical reality described by quantum mechanics. We naively think of such states because we are used to think classically and cannot really wrap our heads around what the concept of "identical particles" really means.
Ok, but what about things like degeneracy pressure then? In some circumstances, like in dying stars, Pauli's exclusion principle really seems to behave like a force in the conventional sense, contrasting the gravitational force and preventing white dwarves from collapsing into a point. How do we reconcile the above described "statistical effect" with this?
What I think is a good way of thinking about this is the following: you are trying to squish a lot of fermions into the same place. However, Pauli's principle dictates a vanishing probability of any pair of them occupying the same position.
The only way to reconcile these two things is that the position distribution of any fermion (say, the $i$-th fermion) must be extremely localised at a point (call it $x_i$), different from all the other points occupied by the other fermions. It is important to note that I just cheated for the sake of clarity here: you cannot talk of any fermion as having an individual identity: any fermion will be very strictly confined in all the $x_i$ positions, provided that all the other fermions are not. The net effect of all this is that the properly antisymmetrized wavefunction of the whole system will be a superposition of lots of very sharp peaks in the high dimensional position space. And it is at this point that Heisenberg's uncertainty comes into play: very peaked distribution in position means very broad distribution in the momentum, which means very high energy, which means that the more you want to squish the fermions together, the more energy you need to provide (that is, classical speaking, the harder you have to "push" them together).
To summarize: due to Pauli's principle the fermions try so hard to not occupy the same positions, that the resulting many-fermion wavefunction describing the joint probabities becomes very peaked, highly increasing the kinetic energy of the state, thus making such states "harder" to reach.
Here (and links therein) is another question discussing this point.
Not the answer you're looking for? Browse other questions tagged quantum-mechanics wavefunction pauli-exclusion-principle or ask your own question.
Is Pauli-repulsion a "force" that is completely separate from the 4 fundamental forces?
What is the formal definition of a system?
How can Pauli's exclusion principle originate forces?
Can two solid objects pass through each other if they are moving sufficiently fast relative to each other?
Does Pauli Exclusion forbid two neutral fermions to occupy the same location in space? | CommonCrawl |
In general, if given equation of any two curves, how to find the shortest distance?
According to me, finding common normal won't work as it isn't necessary for both of them to have one like in case of domain bounded functions.
We can take a more general approach i.e. by assuming points on both the curves, forming the expression for distance between the points and then minimizing it using partial derivatives.
Though the latter approach is reliable but too lengthy and many a times it produces equations quite difficult to solve especially in case of conics. Isn't there a better method?
However, the problem becomes much simpler is you minimize the square of the distance; this will give you the same result.
If the problem is still too difficult, make a grid search (only two parameters in $2D$) and zoom more and more around the minimum. Similar to this, you could make a contour plot of the distance as a function of $x_1$ and $x_2$.
If you have a difficult problem as an example, feel free to post it.
Not the answer you're looking for? Browse other questions tagged calculus differential-geometry curves or ask your own question. | CommonCrawl |
Machine Learning has become a big topic in the IT world nowadays. There are thousands of articles about how ML is changing the way big data is analyzed. High level business decisions are taking place using ML algorithms to make accurate predictions and visualizing trends in the market.
Now, let's ask our selves what exactly is Machine Learning and why a Web Developer would need to learn it?.
Let's start with a simple definition of Machine Learning. As the name suggests, ML is the ability given to a program to learn from experience or adapt to change as humans do, without being explicitly programmed.
Machine Learning as a technology is a body of knowledge comprising a collection of algorithms intended to deal with heterogeneous and always changing data, a characteristic we expect when collecting data from the real world, including data sets with non-linear modeling (e.g., face recognition).
ML algorithms are useful in a variety of applications, as simple as finding a linear model to predict house prices in the real estate, to more complex, nonlinear problems like handwriting and speech recognition in smartphones.
Nevertheless, Machine Learning is not something new. It's historically related to Computational statistics and Computational learning theory. Arthur Samuel coined its name in 1959 while at IBM .
For a computer, to learn from processing large sets of data, it has needed some means to find sense in the data, a method to extract the "knowledge" or to use statistical methods in combination with software algorithms, as we shall see in the further sections.
Problems as simple as taking the mean of a data series or finding the best equation that fit the relation between some parameter "y" as a function of "x," using linear regression on a data set, are considered ML algorithms. The key here is teaching a computer how to calculate the parameters that solve a problem automatically when input data changes.
For a software developer, ML is used to solve certain kind of complex problems using the knowledge of well algorithms and patterns. It is ideally independent of the programming language or development platform. However, there are some famous and influential libraries written in Python, like TensorFlow or scikit that suites in some complex scenarios.
However, I will try to focus in the algorithms and mathematical background in the following sections, as they are the foundations for solving Machine Learning problems and it provides us with the tools to work in any development environment.
Usually, Machine Learning Algorithms are classified in two categories: Supervised and Unsupervised, depending on the need to provide a set of "correct" series of expected outcomes for the algorithm, that is the case of supervised. While if the algorithm can find patterns without any extra parameters or prior training, is considered unsupervised.
Let´s review each of the most relevant algorithms in these categories.
Note: The main academic sources for the following algorithms come from an excellent course taught by Andrew Ng in Stanford University and hosted in Coursera: Machine Learning. Please don't hesitate by subscribing to this course to start learning ML. It is free; there is a paid certificate after completion.
Supervised Learning is the process of inferring a function relation or function from labeled training data (a set of input values and the correct matches or desired output values).
An algorithm in this category can be "trained" with a training data set, taking inputs and the labeled outputs and then find a function between an input and the expected output; thus it can predict results with a new dataset consisting on unseen inputs.
Perhaps the most straightforward algorithm studied in ML courses. Linear Regression as briefly stated before, it is about finding a linear function to model the relationship between two or parameters in a dataset.
Traditionally the function that gives us a desired output from the input set is called a Hypothesis function $h_\theta(x)$, where $\theta$ refers to each feature, a constant or weight that changes the graph form to fit the training data.
We need then to find a hypothesis function. We'll consider a simple case with only a simple input variable x, let's define $h(x) = \theta_0 + \theta_1*x$. This function maps variable y to variable x thus "predicts" any output value $y$ for any value of $x$.
However, how we find the appropriate values of $\theta_0$ and $\theta_1$ based on the training set?.
We start with the mathematical definition of Cost Function, which conceptually is a function that represents how much we get closer or far from the optimal match for a data set when a and b varies (two dimensions), or in other words, how much error we get with a particular selection of values for a and b.
Our task is finding an algorithm that minimizes this function or finding the bottom of the valley in this chart and from there, pick the corresponding values for $\theta_0$ and $\theta_1$.
The algorithm we use to minimize our Cost Function $J(\theta)$ is Gradient Descent, an algorithm that finds the minimum value or valley in the chart by continuously tracking the path "downhill" making small steps at a time (expressed as $\alpha$) until it finds the minimum value, as illustrated in the image below [2:1].
Logistic Regression is used to solve a different kind of problem: Classification. Given a training set, we can develop an algorithm that takes an input and provides a probability as its output, a value between 0 and 1, representing two possible classes, one in the range $0 - 0.5$ and the other $0.5 - 1$.
In this function plot, we can appreciate that many input values go smoothly close to 0 or close to 1, representing an exact match, but some values go in probability zone, for example, $h_theta(x) = 0.75 gives us a probability of 70% that the output is 1.
As with Linear Regression, our problem is finding the feature values for $\theta_0$ and $\theta_1$, that maps into $z = \theta^Tx$ and defines the classification behavior in our hypothesis function.
Multiclass classification is a strategy that allows the use of Logistic Regression (an algorithm that can handle only 1 class) to handle several classes from the same input set.
Thank you so much for reading this article, please take care, and I'll see you next time with a second part, we'll see the algorithms for Neural Networks and Support Vector Machines. Stay tuned!. | CommonCrawl |
Consider the following expression grammar. The semantic rules for expression evaluation are stated next to each grammar production.
Assume the conflicts of this question are resolved using yacc tool and an LALR(1) parser is generated for parsing arithmetic expressions as per the given grammar. Consider an expression $3 \times 2 + 1$. What precedence and associativity properties does the generated parser realize?
the question asks- What precedence and associativity properties does the generated parser realize?
according to me + is having higher precedence than *.
Since yaac prefers shift over reduce and it performs 2+1 first and then multiplied so how + and * have same precedence?
@sushmita LALR parser is SR parser here we use right most derivation...that's why we use right associativity...because we reduced from right ..
All the productions are in same level therefore all have same precedence.
If a grammar has same precedence for 2 different operator (+,*) then it is ambiguous Grammar.
yes i checked with that.Because on reading "num" ,we will go to the state that will reduce E->num. and there are no conflicts,is this correct ?
Doubt:- 3 is look ahead,then you applied E->3 and reduce it to E,after that why is it E->3*?Here stack contains only E* as E->3 is reduced?
Also this E->3 is reduced because there is no SR conflict on reading num?
and yes E->3, this reduction took place because there was no SR conflict in that state where this reduction is done, otherwise, shift move would be taken over reduce by YACC.
Can someone please explain how the preference of S over R is leading to right associativity?
I have made the parsing table from DFA, using which input can be parsed and checked what happens when shift is favoured over reduce.
Equal precedence and Right associativity observed.
@Ayush Upadhyaya I am at state 0 and looking at 3... what shall I do according to your table? There is no reduce move from E-> num.
None of the Options is correct here ... The answer has to be "precedence of + is higher than * " and "both * and + are right associative"
although your answer explains that in certain cases when the grammar poses an RR conflict, the grammar rule which comes before will get a priority, it still is misleading as far as this grammar and question is concerned.
Since the case that you are mentioning here (RR conflict priority decision) doesn't concern the grammar given in question, I think the correct option, according to you, that " precedence of + is higher than * " is simply wrong.
Also after reading your answer, I feel that you don't clearly understand the precedence and associativity concepts. Always, remember, precedence is established first. Once the precedence rules are established, and there comes a case where two operators come one after another and have equal precedence (the operators may be same or different), only then, we apply the associativity rule.
In the last part of your answer, you are using the associativity rule to derive the precedence of + and x which I find to be a reason in concluding the possibility of confusion that you might have regarding associativity and precedence.
Nice catch though regarding RR conflict. | CommonCrawl |
Using quantum Monte Carlo simulations, we compute the participation (Shannon-R\'enyi) entropies for groundstate wave functions of Heisenberg antiferromagnets for one-dimensional (line) subsystems of length $L$ embedded in two-dimensional ($L\times L$) square lattices. We also study the line entropy at finite temperature, i.e. of the diagonal elements of the density matrix, for three-dimensional ($L\times L\times L$) cubic lattices. The breaking of SU(2) symmetry is clearly captured by a universal logarithmic scaling term $l_q\ln L$ in the R\'enyi entropies, in good agreement with the recent field-theory results of Misguish, Pasquier and Oshikawa [arXiv:1607.02465]. We also study the dependence of the log prefactor $l_q$ on the R\'enyi index $q$ for which a transition is detected at $q_c\simeq 1$.
A1 - Luitz, David J.
the R\'enyi index $q$ for which a transition is detected at $q_c\simeq 1$. | CommonCrawl |
Informally, the term infinity is used to mean some infinite number, but this concept falls very far short of a usable definition.
The symbol $\infty$ (supposedly invented by John Wallis) is often used in this context to mean an infinite number.
However, outside of its formal use in the definition of limits its use is strongly discouraged until you know what you're talking about.
The latter result seems wrong when you think of the rule that a negative number square equals a positive one, but remember that infinity is not exactly a number as such.
The term ad infinitum can often be found in early texts. It is Latin for to infinity.
The concept of infinity has bothered scientists, mathematicians and philosophers since the time of Aristotle.
The symbol $\infty$ for infinity was introduced by John Wallis in the $17$th century.
It was Georg Cantor in the $1870$s who finally made the bold step of positing the actual existence of infinite sets as mathematical objects which paved the way towards a proper understanding of infinity. | CommonCrawl |
Abstract: Consider the relativistic Vlasov-Maxwell system with initial data of unrestricted size. In the two dimensional and the two and a half dimensional cases, Glassey-Schaeffer (1997, 1998, 1998) proved that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the 2D and the $2\frac 12$D cases, we prove the global existence, uniqueness and regularity for solutions arising from this class of initial data. To this end we use Strichartz estimates and prove that suitable moments of the solution remain bounded. Moreover, we obtain a slight improvement of the temporal growth of the $L^\infty_x$ norms of the electromagnetic fields compared to Glassey-Schaeffer. | CommonCrawl |
At the beginning of this section, we claimed that breadth-first search finds the distance to each reachable vertex in a graph $G=(V,E)$ from a given source vertex $s \in V$ . Define the shortest-path distance $\delta(s,v)$ from $s$ to $v$ as the minimum number of edges in any path from vertex $s$ to vertex $v$; if there is no path from $s$ to $v$, then $\delta(s,v)=\infty$. We call a path of length $\delta(s,v)$ from $s$ to $v$ a shortest path from $s$ to $v$.
So for many months, I was content with the fact that for unweighted path, BFS gives shortest path. But today I came across problem asking whether breadth first search gives minimum spanning tree for unweighted graph. I was like, BFS gives shortest path not the minimum spanning tree. And to my surprise I was wrong. Somehow, stupidly, I assumed what CLRS stated was the only connection among minimum spanning tree, shortest path, depth first search and breadth first search, because I was subconsciously thinking that its not given in CLRS (in any of four sections), then it should not be the case. I did not give any extra thought for evaluating any possible connection. But now I want to know what all are the connection.
I have done quick google and found many links, some of which are:1, 2, 3.
Breadth first search gives both minimum spanning tree and shortest path tree.
Depth first search gives only minimum spanning tree but not the shortest path tree.
I know for weighted graph MST and SPT are not same. But are they same for unweighted graph? Somehow I feel no, as otherwise point 2 will be wrong, and DFS would have given both MST and SPT for unweighted graph. However I am not able to come up with the unweighted graph for which MST and SPT are different.
MSTs given by BFS and DFS on given unweighted graph may be different, its just that the number of edges contained in them will be the same.
Which of above points are correct and which are wrong?
In fact, a MST is a subset of the edges building a tree (thus without cycle) using the minimum edge weights. It has exactly N-1 edges for a N nodes graph.
It may exist several different MST for the same graph. For an unweighted graph, which is in other way a graph with weight 1 on every edges, the total weight of any spanning tree is N-1. Thus any spanning tree is a MST.
When you use BFS or DFS to explore a graph, you build an exploration tree, which is the shortest path tree for BFS. And if the graph is unweighted, you can say this tree is a MST, yes so 1. and 2. are true.
3. is true, MST and SPT are different. Just make BFS and DFS on a 3-clique (3 nodes connected to each other) to have an example.
4. is true but remember that "minimum" adds no information to any spanning tree of an unweighted graph. And also that a N nodes tree has exactly N-1 egdes.
(Q1,2) Thus, as long as we have a tree connecting all vertices and thus having n-1 edges, we are ending up with MST, whatever the procedure is followed.
(Q1,2) As can be seen from above, BFS gives SPT which also happen to be MST. But DFS gives MST which may not be a SPT.
Not the answer you're looking for? Browse other questions tagged graphs graph-theory shortest-path minimum-spanning-tree or ask your own question.
Why can't DFS be used to find shortest paths in unweighted graphs?
How does DFS produce MST and All pairs shortest paths in unweighted graphs?
Shortest path tree from each vertex implies a unique MST? | CommonCrawl |
Abstract : For a real number $0<\lambda<2$, we introduce a transformation $T_\lambda$ naturally associated to expansion in $\lambda$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of $T_\lambda$ provides an algorithm to expand any positive real number in lambda-continued fraction. We prove the conjugacy between $T_\lambda$ and some beta-shift, $\beta>1$. Some properties of the map $\lambda\mapsto\beta(\lambda)$ are established: It is increasing and continuous from ]0, 2[ onto ]1,\infty[ but non-analytic. | CommonCrawl |
You are on an $n \times m$ grid where each square on the grid has a digit on it. From a given square that has digit $k$ on it, a Move consists of jumping exactly $k$ squares in one of the four cardinal directions. A move cannot go beyond the edges of the grid; it does not wrap. What is the minimum number of moves required to get from the top-left corner to the bottom-right corner?
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line of input contains two space-separated integers $n$ and $m$ ($1 \le n, m \le 500$), indicating the size of the grid. It is guaranteed that at least one of $n$ and $m$ is greater than $1$.
The next $n$ lines will each consist of $m$ digits, with no spaces, indicating the $n \times m$ grid. Each digit is between 0 and 9, inclusive.
The top-left corner of the grid will be the square corresponding to the first character in the first line of the test case. The bottom-right corner of the grid will be the square corresponding to the last character in the last line of the test case.
Output a single integer on a line by itself representing the minimum number of moves required to get from the top-left corner of the grid to the bottom-right. If it isn't possible, output -1. | CommonCrawl |
For questions about mathematical induction, a method of mathematical proof. Mathematical induction generally proceeds by proving a statement for some integer, called the base case, and then proving that if it holds for one integer then it holds for the next integer. This tag is primarily meant for questions about induction over natural numbers but is also appropriate for other kinds of induction such as transfinite, structural, double, backwards, etc.
Mathematical induction is a form of deductive reasoning. Its most common use is induction over well-ordered sets, such as natural numbers or ordinals. While induction can be expanded to class relations which are well-founded, this tag is aimed mostly at questions about induction over natural numbers.
In general use, induction means inference from the particular to the general. This is used in terms such as inductive reasoning, which involves making an inference about the unknown based on some known sample. Mathematical induction is not true induction in this sense, but is rather a form of proof.
First prove the statement for the base case, which is usually $n=0$ or $n=1$.
Next, assume that the statement is true for an input $n$, and prove that it is true for the input $n+1$.
The following variant goes without a base case: Assuming the statement is true for all $n\in\mathbb N$ with $n < N$, prove that is true for $N$, too. This has to be done for all $N\in\mathbb N$. | CommonCrawl |
Our sad tale begins with a tight clique of friends. Together they went on a trip to the picturesque country of Molvania. During their stay, various events which are too horrible to mention occurred. The net result was that the last evening of the trip ended with a momentous exchange of "I never want to see you again!"s. A quick calculation tells you it may have been said almost $50$ million times!
Back home in Scandinavia, our group of ex-friends realize that they haven't split the costs incurred during the trip evenly. Some people may be out several thousand crowns. Settling the debts turns out to be a bit more problematic than it ought to be, as many in the group no longer wish to speak to one another, and even less to give each other money.
Naturally, you want to help out, so you ask each person to tell you how much money she owes or is owed, and whom she is still friends with. Given this information, you're sure you can figure out if it's possible for everyone to get even, and with money only being given between persons who are still friends.
The first line contains two integers, $n$ ($2 \leq n \leq 10\, 000$), and $m$ ($0 \le m \leq 50\, 000$), the number of friends and the number of remaining friendships. The friends are named $0, 1, \ldots , n-1$. Then $n$ lines follow, each containing an integer $o$ ($-10\, 000 \le o \le 10\, 000)$ indicating how much each person owes (or is owed if $o <0$). The first of those lines gives the balance of person $0$, the second line the balance of person $1$, and so on. The sum of these values is zero.
After this comes $m$ lines giving the remaining friendships, each line containing two integers $x$, $y$ ($0 \le x < y \le n-1$) indicating that persons $x$ and $y$ are still friends.
Your output should consist of a single line saying "POSSIBLE" or "IMPOSSIBLE". | CommonCrawl |
If twice the son's age in years is added to the father's age, the sum is 70. But if the twice the father's age is added to the son's age, the sum is 95. Find the ages of father and son.
Strategy : Given word problem can be solved by converting it into a set of simultaneous equations. Solution : Let the age of son be X and that of the father be Y. then Equation 1: $$2X + Y = 70$$ by Condition 1 - twice the son's age in years is added to the father's age, the sum is 70 Equation 2: $$X + 2Y = 95$$ by Condition 2 - twice the father's age is added to the son's age, the sum is 95 Let's rearrange the equation 1 to get the value of Y in terms of X. $$ Y = 70 - 2X$$ Put this value of Y in Equation 2. $$X + 2 \times (70 - 2X) = 95$$ $$X + 140 - 4X = 95$$ $$140 - 3X = 95$$ $$140 - 95 = 3X$$ $$45=3X$$ $$X=15$$ ; Put this value of X in Equation 1 we get, $$2 \times 15 + Y = 70$$ $$ 30 + Y =70$$ $$ Y = 70-30$$ $$ Y = 40$$ Hence the age of son and father are 15 and 40 simultaneously.
Air at 100 kPa, 290 K enters an ideal Otto cycle. The initial volume is 600 cm3. The compression ratio is 9.5, and the Temperature at the end of an isentropic expansion is 800 K. Find 1) Highest Temperature 2) Amount of heat added Use constant specific heat. Properties of air at room temperature: $$c_p = 1.005\times 10^3 J/kgK, c_v = 0.718 \times 10^3 J/kgK, k = 1.4, R = 0.287 \times 10^3 J/kgK. $$.
A stationary block of mass 2 kg placed on a long frictionless horizontal table is pulled horizontally by a constant force F. It is found to move 10 m in the first two seconds. Find the magnitude of F.
This question belongs to Newton's Laws of motion and Kinematic Equations. Given : mass (M) = 2 Kg, distance moved (s) = 10 m and time (t) = 2 sec. To Find : The magnitude of force required (F) Strategy : We have to find the force (F) required to pull this block. As we know from Newtons Second law that $$ F = M \times a $$, where $$M$$ is mass and $$a$$ is acceleration. So in order to find force ($$F$$) we need mass ($$M$$) and acceleration ($$a$$) of the block. We already know the mass ($$M$$) of the block. Hence all we have to do is find acceleration ($$a$$) of the block using distance ($$s$$) moved by the block in time ($$t$$). From second equation of motion $$s = u\times t + 1/2 \times a\times t^2$$, where u is the initial velocity. As given in the problem, initially the block is stationary which implies $$u= 0 $$. Hence we can find $$a$$ using this equation and put it back into $$ F = M \times a $$ to get the required force. Calculation : $$s = u\times t + 1/2 \times a\times t^2$$ $$10 = 0\times 2 + 1/2 \times a\times 2^2$$ $$10 = 2 \times a$$ $$a = 5 m/sec^2 $$ $$ F = M \times a $$ $$ F = 2 \times 5 $$ $$ F = 10 N$$. Hence the force of 10N will be required to pull the block. | CommonCrawl |
In a previous question (Calabi-Yau manifolds and compactification of extra dimensions in M-theory), I was told that the $G(2)$ lattice can be used to compactify the extra 7 dimensions of M-theory and preserve exactly $\mathcal N=1$ supersymmetry.
However, since there is only 1 $G(2)$ lattice, there should be only 1 4-dimensional M-theory. Then, why is there such a huge fuss about the M-theory landscape?
It's not a "$G(2)$ lattice" one has to compactify the M-theoretical dimensions upon (after all, the $G_2$ lattice is 2-dimensional); it's the $G_2$ holonomy manifolds. There are lots of different topologies of these seven-dimensional manifolds. They're analogous to the Calabi-Yau manifolds but don't allow one to use the machinery of complex numbers.
Not the answer you're looking for? Browse other questions tagged string-theory supersymmetry ads-cft string-theory-landscape calabi-yau or ask your own question.
Why F-theory picks Calabi-Yau manifolds as backgrounds?
What keeps strings in their proper "shape" despite their enormous inherent tension?
How does one actually apply the M-theory/heterotic duality "fiberwise"?
Is this a correct non-technical description of mirror symmetry for Calabi-Yau manifolds arising from string theory? | CommonCrawl |
number of orbits of a group: is map well-defined and surjective?
Let $G$ be a finite group acting on a finite set $X$. Let $m$ be a number of orbits of $G$ on $X$ and $M$ be the number of orbits of $G$ on $X\times X$. Show that $m^2\le M$ with equality if and only if G acts trivialy on $X$.
I need your help to solve this problem. Thanks.
I want to check that this map is well-defined and surjective. | CommonCrawl |
We show that the Novikov–Shubin invariant of an element of the integral group ring of the lamplighter group $\mathbf Z_2 \wr \mathbf Z$ can be irrational. This disproves a conjecture of Lott and Lück. Furthermore we show that every positive real number is equal to the Novikov–Shubin invariant of some element of the real group ring of $\mathbf Z_2 \wr \mathbf Z$. Finally we show that the $l^2$-Betti number of a matrix over the integral group ring of the group $\mathbf Z_p \wr \mathbf Z$, where $p$ is a natural number greater than $1$, can be irrational. As such the groups $\mathbf Z_p \wr \mathbf Z$ become the simplest known examples which give rise to irrational $l^2$-Betti numbers. | CommonCrawl |
Abstract: Eilenberg-type correspondences, relating varieties of languages (e.g. of finite words, infinite words, or trees) to pseudovarieties of finite algebras, form the backbone of algebraic language theory. Numerous such correspondences are known in the literature. We demonstrate that they all arise from the same recipe: one models languages and the algebras recognizing them by monads on an algebraic category, and applies a Stone-type duality. Our main contribution is a variety theorem that covers e.g. Wilke's and Pin's work on $\infty$-languages, the variety theorem for cost functions of Daviaud, Kuperberg, and Pin, and unifies the two previous categorical approaches of Bojańczyk and of Adámek et al. In addition we derive a number of new results, including an extension of the local variety theorem of Gehrke, Grigorieff, and Pin from finite to infinite words. | CommonCrawl |
Given an $n$-vertex weighted tree with structural diameter $S$ and a subset of $m$ vertices, we present a technique to compute a corresponding $m \times m$ Gram matrix of the pseudoinverse of the graph Laplacian in $O(n+ m^2 + m S)$ time. We discuss the application of this technique to fast label prediction on a generic graph. We approximate the graph with a spanning tree and then we predict with the kernel perceptron. We address the approximation of the graph with either a minimum spanning tree or a shortest path tree. The fast computation of the pseudoinverse enables us to address prediction problems on large graphs. To this end we present experiments on two web-spam classification tasks, one of which includes a graph with 400,000 nodes and more than 10,000,000 edges. The results indicate that the accuracy of our technique is competitive with previous methods using the full graph information. | CommonCrawl |
I have an integer linear program (ILP) with some variables $x_i$ that are intended to represent boolean values. The $x_i$'s are constrained to be integers and to hold either 0 or 1 ($0 \le x_i \le 1$).
I want to express boolean operations on these 0/1-valued variables, using linear constraints. How can I do this?
More specifically, I want to set $y_1 = x_1 \land x_2$ (boolean AND), $y_2 = x_1 \lor x_2$ (boolean OR), and $y_3 = \neg x_1$ (boolean NOT). I am using the obvious interpretation of 0/1 as Boolean values: 0 = false, 1 = true. How do I write ILP constraints to ensure that the $y_i$'s are related to the $x_i$'s as desired?
Logical OR: Use the linear constraints $y_2 \le x_1 + x_2$, $y_2 \ge x_1$, $y_2 \ge x_2$, $0 \le y_2 \le 1$, where $y_2$ is constrained to be an integer.
Logical NOT: Use $y_3 = 1-x_1$.
Logical implication: To express $y_4 = (x_1 \Rightarrow x_2)$ (i.e., $y_4 = \neg x_1 \lor x_2$), we can adapt the construction for logical OR. In particular, use the linear constraints $y_4 \le 1-x_1 + x_2$, $y_4 \ge 1-x_1$, $y_4 \ge x_2$, $0 \le y_4 \le 1$, where $y_4$ is constrained to be an integer.
Forced logical implication: To express that $x_1 \Rightarrow x_2$ must hold, simply use the linear constraint $x_1 \le x_2$ (assuming that $x_1$ and $x_2$ are already constrained to boolean values).
XOR: To express $y_5 = x_1 \oplus x_2$ (the exclusive-or of $x_1$ and $x_2$), use linear inequalities $y_5 \le x_1 + x_2$, $y_5 \ge x_1-x_2$, $y_5 \ge x_2-x_1$, $y_5 \le 2-x_1-x_2$, $0 \le y_5 \le 1$, where $y_5$ is constrained to be an integer.
Cast to boolean (version 1): Suppose you have an integer variable $x$, and you want to define $y$ so that $y=1$ if $x \ne 0$ and $y=0$ if $x=0$. If you additionally know that $0 \le x \le U$, then you can use the linear inequalities $0 \le y \le 1$, $y \le x$, $x \le Uy$; however, this only works if you know an upper and lower bound on $x$. Or, if you know that $|x| \le U$ (that is, $-U \le x \le U$) for some constant $U$, then you can use the method described here. This is only applicable if you know an upper bound on $|x|$.
Cast to boolean (version 2): Let's consider the same goal, but now we don't know an upper bound on $x$. However, assume we do know that $x \ge 0$. Here's how you might be able to express that constraint in a linear system. First, introduce a new integer variable $t$. Add inequalities $0 \le y \le 1$, $y \le x$, $t=x-y$. Then, choose the objective function so that you minimize $t$. This only works if you didn't already have an objective function. If you have $n$ non-negative integer variables $x_1,\dots,x_n$ and you want to cast all of them to booleans, so that $y_i=1$ if $x_i\ge 1$ and $y_i=0$ if $x_i=0$, then you can introduce $n$ variables $t_1,\dots,t_n$ with inequalities $0 \le y_i \le 1$, $y_i \le x_i$, $t_i=x_i-y_i$ and define the objective function to minimize $t_1+\dots + t_n$. Again, this only works nothing else needs to define an objective function (if, apart from the casts to boolean, you were planning to just check the feasibility of the resulting ILP, not try to minimize/maximize some function of the variables).
For some excellent practice problems and worked examples, I recommend Formulating Integer Linear Programs: A Rogues' Gallery.
For NOT, no such improvement is available.
Not the answer you're looking for? Browse other questions tagged linear-programming integer-programming or ask your own question.
Why is linear programming in P but integer programming NP-hard?
Does every NP problem have a poly-sized ILP formulation?
Is 0-1 integer linear programming NP-hard when $c^T$ is the all-ones vector?
How to solve an ILP problem with conditions in an objective function? | CommonCrawl |
We find the capacity pre-log of a temporally correlated Rayleigh block-fading single-input multiple-output (SIMO) channel in the noncoherent setting. It is well known that for blocklength $L$ and rank of the channel covariance matrix equal to $Q$, the capacity pre-log in the single-input single-output (SISO) case is given by $1-Q/L$. Here, $Q/L$ can be interpreted as the pre-log penalty incurred by channel uncertainty. Our main result reveals that, by adding only one receive antenna, this penalty can be reduced to $1/L$ and can, hence, be made to vanish for the blocklength $L\to\infty$, even if $Q/L$ remains constant as $L\to\infty$. Intuitively, even though the SISO channels between the transmit antenna and the two receive antennas are statistically independent, the transmit signal induces enough statistical dependence between the corresponding receive signals for the second receive antenna to be able to resolve the uncertainty associated with the first receive antenna's channel and thereby make the overall system appear coherent. The proof of our main theorem is based on a deep result from algebraic geometry known as Hironaka's Theorem on the Resolution of Singularities. | CommonCrawl |
Recall from The Squares of Riemann-Stieltjes Integrable Functions with Increasing Integrators page that if $f$ is a function defined on $[a, b]$ and $\alpha$ is an increasing function on $[a, b]$ then if $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ then $f^2$ is also Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$.
We will now use this important theorem to show that if $f$ and $g$ are both functions defined on $[a, b]$, $\alpha$ is increasing on $[a, b]$, and $f$ and $g$ are Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ then their product $fg$ is also Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$.
Theorem 1: Let $f$ and $g$ both be functions defined on $[a, b]$ and let $\alpha$ be an increasing function on $[a, b]$. If $f$ and $g$ are Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ then $fg$ is also Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$.
Since $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ we have that $\int_a^b [f(x)]^2 \: d \alpha (x)$ exists. Similarly, since $g$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ we have that $\int_a^b [g(x)]^2 \: d \alpha (x)$.
Furthermore, from the Linearity of the Integrand of Riemann-Stieltjes Integrals page we see that the sum $f + g$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$, so $(f + g)^2$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$. | CommonCrawl |
This is a short guide how to format citations and the bibliography in a manuscript for Progress in Disaster Science. For a complete guide how to prepare your manuscript refer to the journal's instructions to authors.
McDonald F. Watching the players at the climate poker table. Nature 2011;480:293.
Turecek R, Trussell LO. Presynaptic glycine receptors enhance transmitter release at a mammalian central synapse. Nature 2001;411:587–90.
Deuss A, Irving JCE, Woodhouse JH. Regional variation of inner core anisotropy from seismic normal mode observations. Science 2010;328:1018–20.
Wager TD, Rilling JK, Smith EE, Sokolik A, Casey KL, Davidson RJ, et al. Placebo-induced changes in FMRI in the anticipation and experience of pain. Science 2004;303:1162–7.
Stephans RA. System Safety for the 21 st Century. Hoboken, NJ: John Wiley & Sons, Inc.; 2004.
Pickel G, Sammet K, editors. Transformations of Religiosity: Religion and Religiosity in Eastern Europe 1989 – 2010. Wiesbaden: VS Verlag für Sozialwissenschaften; 2012.
Uohashi K. Harmonic Maps Relative to $$\alpha $$ α -Connections. In: Nielsen F, editor. Geometric Theory of Information, Cham: Springer International Publishing; 2014, p. 81–96.
Sometimes references to web sites should appear directly in the text rather than in the bibliography. Refer to the Instructions to authors for Progress in Disaster Science.
Andrew E. Inoculating Against Science Denial. IFLScience 2015. https://www.iflscience.com/health-and-medicine/inoculating-against-science-denial/ (accessed October 30, 2018).
Government Accountability Office. Tax Systems Modernization: Unmanaged Risks Threaten Success. Washington, DC: U.S. Government Printing Office; 1995.
Xiong Y. Immuno-Magnetic T Cell Depletion for Allogeneic Hematological Stem Cell Transplantation. Doctoral dissertation. Ohio State University, 2008.
Ivory D, Protess B, Palmer G. In American Towns, Pumping Private Profit From Public Works. New York Times 2016:A1. | CommonCrawl |
Calculates the scattering from a barbell-shaped cylinder. Like `capped-cylinder`, this is a sphereocylinder with spherical end caps that have a radius larger than that of the cylinder, but with the center of the end cap radius lying outside of the cylinder. See the diagram for the details of the geometry and restrictions on parameter values.
The $\left<\ldots\right>$ brackets denote an average of the structure over all orientations. $\left<A^2(q,\alpha)\right>$ is then the form factor, $P(q)$. The scale factor is equivalent to the volume fraction of cylinders, each of volume, $V$. Contrast $\Delta\rho$ is the difference of scattering length densities of the cylinder and the surrounding solvent.
.. note:: The requirement that $R \geq r$ is not enforced in the model! It is up to you to restrict this during analysis.
Definition of the angles for oriented 2D barbells. | CommonCrawl |
Abstract: We address the Riemann and Cauchy problems for systems of $n$ conservation laws in $m$ unknowns which are subject to $m-n$ constraints ($m\geq n$). Such constrained systems generalize systems of conservation laws in standard form to include various examples of conservation laws in Physics and Engineering beyond gas dynamics, e.g., multi-phase flow in porous media. We prove local well-posedness of the Riemann problem and global existence of the Cauchy problem for initial data with sufficiently small total variation, in one spatial dimension. The key to our existence theory is to generalize the $m\times n$ systems of constrained conservation laws to $n\times n$ systems of conservation laws with states taking values in an $n$-dimensional manifold and to extend Lax's theory for local existence as well as Glimm's random choice method to our geometric framework. Our resulting existence theory allows for the accumulation function to be non-invertible across hypersurfaces. | CommonCrawl |
of $f$ at $x_0 \in [0,1]$ .
For any $\epsilon \gt 0$,for any $\delta \gt 0$ such that for all $x \in (0,1)$ and $0 \lt|x-x_0| \lt \delta$, one has $|f(x) - L| \lt \epsilon$ .
This definition is incorrect because for any $\epsilon \gt 0$ there exists some $\delta \gt 0$ that is small enough. It can't be any delta. Is this the only reason why this definition is not valid?
and the other part of the definition is correct.
Your definition implies the known definition but the converse is not true.
Not the answer you're looking for? Browse other questions tagged limits definition epsilon-delta or ask your own question.
Is my understanding of a limit correct?
Is there any surjective property in $\epsilon-\delta$ relation in the definition of limit?
How to prove that a limit is incorrect using epsilon delta definition of a limit?
equivalent definition of limit of a function? | CommonCrawl |
Аннотация: The paper is concerned with the large time behavior of solutions of the heat equation on a thin two-layer domain. Such systems may arise from modeling thermal emission (as a result of chemical reaction) and heat transfer between two thin films. It is shown that every solution converges as $t\to+\infty$ to a single equilibrium point. | CommonCrawl |
Abstract: We consider the problem of the reduction of unitary irreducible representations of the generalized Poincaré groups $\mathscr P(1,n)$ with respect to their subgroups $\mathscr P(1, n-k)$. We find the explicit form of the unitary operator that relates the canonical basis of the representation to the $\mathscr P(1, n-k)$-basis. The action of the generators in the $\mathscr P(1, n-k)$-basis is given explicitly. The case of the inhomogeneous de Sitter group is considered in detail. | CommonCrawl |
A maximal subgroup of $\GL(2,\F_p)$ that fixes a one-dimensional subspace of $\F_p^2$ is called a Borel subgroup. Every Borel subgroup is conjugate to the subgroup of upper triangular matrices.
Subgroup labels containing the letter B identify a subgroup of $\GL(2,\F_p)$ that lies in the Borel subgroup of upper triangular matrices but is not contained in the subgroup of diagonal matrices; these are precisely the subgroups of a Borel subgroup that contain an element of order $p$.
where $a$ and $b$ are minimally chosen positive integers and $r$ is the least positive integer generating $(\Z/p\Z)^\times\simeq \F_p^\times$, as defined in [MR:3482279] . | CommonCrawl |
Abstract: We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued, pairwise orthogonal vectors. Such decompositions do not generally exist, but we show that some symmetric tensor decomposition problems can be converted to orthogonal problems following the whitening procedure proposed by Anandkumar et al. (2012). If an orthogonal decomposition of an $m$-way $n$-dimensional symmetric tensor exists, we propose a novel method to compute it that reduces to an $n \times n$ symmetric matrix eigenproblem. We provide numerical results demonstrating the effectiveness of the method. | CommonCrawl |
I'm running an experiment and am suspecting that competitive people may respond differently to experimental treatments compared to non-competitive people.
"My friends would describe me as a competitive person."
"I would describe myself as a competitive person."
"Even when there is no monetary reward, I will seek to surpass others when doing a task."
Would these be good measures of competitiveness? What other questions could I use?
Despite its relevance to a wide variety of situations, competitiveness is a personality characteristic that has not been widely studied. Although some research in need for achievement, sports psychology, experimental social psychology, and personality assessment has approached the topic of competitiveness, few studies have provided a clear definition of the construct or a psychometrically-sound way of measuring it. This paper clarifies the conceptual definition of competitiveness and introduces a 20-item scale called the Competitiveness Index. Study 1 focuses on the validity and reliability of the measure; Study 2 reports the results of an exploratory factor analysis of the Competitiveness Index which yielded three factors: (a) Emotion, (b) Argument, and (c) Games.
But it's 1990's paper with only ~100 citations or so, so not a blockbuster. You should also look at which papers cite it, to see if anything better has been developed.
Little research has been devoted to the study of competitiveness as a personality trait. The Competitiveness Index (CI), developed by Smither and Houston (in press, Educational and Psychological Measurement), is a 20-item structured personality instrument that responds to this problem. This study explored the validity of the CI by investigating both the internal and external validity of the measure. The CI was completed by 255 subjects (158 nurses and 97 attorneys). A confirmatory factor analysis was conducted to assess the stability of the CI's factor structure across different population samples. In addition, a logistic discriminant analysis was performed to evaluate the CI's ability to differentiate between individuals in a competitive occupation (attorney) and a less competitive occupation (staff nurse). The results provide support for both the internal and external validity of the CI. The findings from the confirmatory factor analysis indicate that the factor structure of the CI is consistent across different population samples. The results of the discriminant analysis demonstrate that the CI can differentiate between individuals in more and less competitive occupations. Future research directions and possible applications of the CI are discussed.
Competitiveness. To measure competitiveness, we modified the competitiveness index scale suggested by Houston and colleagues (see Houston, Farese, & La Du, 1992; Smither & Houston, 1992) so that we could capture state competitive mindset. The scale included the following items: 1) Right now, I think that competing against an opponent would be enjoyable and 2) Right now, I think that keeping score is important when playing games. The two items were correlated at r=.73, p<.001, which corresponds to $\alpha$=.84 (M=4.95, SD=1.42).
Good answer above by Fizz.
Not the answer you're looking for? Browse other questions tagged measurement survey or ask your own question. | CommonCrawl |
Zarebnia, M., Aghili, M. (2017). An approximation to the solution of Benjamin-Bona-Mahony-Burgers equation. Computational Methods for Differential Equations, 5(4), 301-309.
Mohammad Zarebnia; Maryam Aghili. "An approximation to the solution of Benjamin-Bona-Mahony-Burgers equation". Computational Methods for Differential Equations, 5, 4, 2017, 301-309.
Zarebnia, M., Aghili, M. (2017). 'An approximation to the solution of Benjamin-Bona-Mahony-Burgers equation', Computational Methods for Differential Equations, 5(4), pp. 301-309.
Zarebnia, M., Aghili, M. An approximation to the solution of Benjamin-Bona-Mahony-Burgers equation. Computational Methods for Differential Equations, 2017; 5(4): 301-309.
In this paper, numerical solution of the Benjamin-Bona-Mahony-Burgers (BBMB) equation is obtained by using the mesh-free method based on the collocation method with radial basis functions (RBFs). Stability analysis of the method is discussed. The method is applied to several examples and accuracy of the method is tested in terms of $L_2$ and $L_\infty$ error norms. | CommonCrawl |
Add up all the numbers from 1 to 100.
The class got busy calculating. But not Gauss. After a few minutes, while most students were just beginning to struggle with the teens, Gauss was just finishing with writing down his final answer.
When he brought it to submit it, his teacher was furious with him for not taking the assignment seriously (and for interrupting his nap behind the desk).
We'll look to see how Gauss got his answer so quickly.
There's an interesting trick for summing consecutive numbers together. It's easy enough that you can teach it to children in elementary school. We'll show a special case, then generalize it.
We've listed the sum twice, once forwards and once backwards. So we won't get the answer, but instead we'll get twice the answer.
And that is just $5 \times 6 = 30$. But wait. Since we've listed the sum twice, our answer, $30$, is actually twice what it needs to be. So to get the value of the original sum, we just cut it in half: $15$.
Adding vertically, how much does each column give you? How many columns of that same number do you get? Remember that you're adding the sum twice, so you will need to half the result of this! | CommonCrawl |
I've recently stumbled upon this very nice interactive visualization of eigenvectors of two-dimensional matrices, and how powers $A^k$ act on various vectors.
How can this sort of visualization be realized with Mathematica, leveraging its dynamical capabilities?
The following is an attempt to recreate a similar sort of interactive visualization, showing the eigenvectors (when real), and how the various points of the unit circle are transformed by the matrix.
The matrix can be chosen by moving its two column vectors using the mouse. I used EventHandler for this, instead of Locators, for greater customizability and a more natural look.
To ease code readability and modularity, the components of the graphical object are defined separately in a private context, and injected into the final DynamicModule object.
Here, the blue arrow corresponds to a unit vector (which traces out a circle), and the green arrow corresponds to the transformed unit vector (which traces out an ellipse). If you can make the blue and green arrows parallel to each other, then the green arrow corresponds to an eigenvector of A.
The slider for n determines how many iterates of $\mathbf A^n\mathbf x$ to take.
Not the answer you're looking for? Browse other questions tagged graphics dynamic linear-algebra visualization eigenvalues or ask your own question.
How to translate interactive graphics from Mathematica to standard HTML+SVG?
Why can I edit only one character of the output of a Dynamic expression?
What if do NOT want Mathematica to normalize eigenvectors with Eigenvectors[N[matrix]]? | CommonCrawl |
Iron carbonyl-mediated Michael homologous reactions of gamma-alkoxy alkenones.
The reaction of $\gamma$-benzyloxy-$\alpha,\beta$-unsaturated ketones with diiron nonacarbonyl afforded the corresponding $\eta\sp2$-iron tetracarbonyl alkene complexes selectively, without perceptible formation of $\eta\sp4$-iron tricarbonyl complexes. In the presence of boron trifluoride-etherate, these complexes formed $\eta\sp3$-allyl tetracarbonyliron cation intermediates, which reacted with silyl enol ethers, silyl ketene acetals, allyltributyltin, and an electron rich arene to afford 59a-d, 59e, 59f, and 59g, respectively. The regiochemistry of the reactions was exclusively $\gamma$- with respect to the carbonyl, and the geometrical stability of the allyl tetracarbonyliron cation intermediates allowed retention of configuration of the double bond. The substrates, $\gamma$-benzyloxy-$\alpha,\beta$-unsaturated ketones (55a-c), were prepared from propargyl alcohols, in four steps. Attempts were made to apply these homologous Michael reactions to the synthesis of the ring system of naturally occurring pyrenotides by preparing diketoesters 59b$\sp\prime$ from $\gamma$-benzyloxy-$\alpha,\beta$-unsaturated ketone (55b) and an appropriate silyl enol ether. The route, employing a silyl enol ether containing an alkynoate function, namely 69, failed to give a coupling product. Silyl enol ether (67), containing a Z-substituted alkenoate function failed to react in the intended manner. Trimethylsilyl enol ether (61), bearing an E-substituted alkenoate function, did react to give the expected homo-Michael adduct (59b), albeit in low yield. Further investigation of this transformation, and its application in making natural products, is discussed.Dept. of Chemistry and Biochemistry. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1994 .Z475. Source: Masters Abstracts International, Volume: 34-02, page: 0767. Adviser: James R. Green. Thesis (M.Sc.)--University of Windsor (Canada), 1994.
Zhou, Tianhao., "Iron carbonyl-mediated Michael homologous reactions of gamma-alkoxy alkenones." (1994). Electronic Theses and Dissertations. 2519. | CommonCrawl |
You are given an array that contains $n$ positive integers.
Your task is to divide the array into $k$ subarrays so that the maximum sum in a subarray is a small as possible.
The first input line contains two integers $n$ and $k$: the size of the array and the number of subarrays in the division.
The next line contains $n$ integers $x_1,x_2,\ldots,x_n$: the contents of the array.
Print one integer: the maximum sum in a subarray in the optimal division.
Explanation: An optimal division is $[2,4],,[3,5]$ where the sums of the subarrays are $6,7,8$. The largest sum is the last sum $8$. | CommonCrawl |
In the path integral approach to gauge theory, observables are gauge invariant functions on the space $\mathcal A$ of a $G$-connections on $E$, where $G$ denotes the structure group and $E$ the fiber bundle. Therefore, an observable $f$ is a function on the space $\mathcal A / \mathcal G$, of connections modulo gauge transformations.
As a result, vacuum expectation values are no longer defined as integrals with Lebesgue measure $ \mathcal A$, but instead with a Lebesgue measure on $ \mathcal A/ \mathcal G$. We obtain this measure by pushing forward the Lebesgue measure on $ \mathcal A$ by the map $ \mathcal A \to \mathcal A/ \mathcal G$ that sends each connection to its gauge equivalence class, and then $ A$ denotes a gauge equivalence class of connections in the integral.
The simplest example of an observable in gauge theory are Wilson loops. | CommonCrawl |
Pyramidal Cells are present in the cerebral cortex and the hippocampus. They exhibit non linear summation of inputs, with dendritic compartments acting as individual subunits capable of producing their own spikes. These dendrites then project to the cell body, where all dendritic signals are summed and integrated to determine the spiking behavior of the Pyramidal Cell body. Thus, Pyramidal Cells are multi-subunit structures, capable of computations which are far more complex than those of a linear point neuron. Properties of Pyramidal Cells useful for modeling are outlined below. Note that there are several different types of pyramidal cell depending on which region of the brain one is considering. Although they are expected to behave similarly, they do have some differences. It is for this reason that information will be given for individual pyramidal cell types.
All pyramidal cells contain the same basic structure. They are composed of a pyramidal shaped cell body, and a single, heavily branching axon projecting from the base.
The dendrites of a pyramidal neuron can be divided into two domains: basal and apical.
The basal dendritic tree is composed of 3-5 primary dendrites. Each of these divides to form branches of progressively thinning length.
The apical tree, which also splits several times, ends in a largely branched section known as the apical tuft.
The dendrites of pyramidal cells are covered with tiny branches known as dendritic spines. As one moves distal to the soma, the number of spines increases. These spines increase the surface area, and are believed to be where the majority of synapses arrive at the dendrites.
From a modeling perspective, both basal and apical branches contain proximal, medial, and distal compartments. These can be viewed as individual computational subunits. In addition, CA1 pyramidal neurons contain oblique medial and oblique distal branches, which arise from the medial apical and medial distal dendrites respectively.
Leads to an EPSP that is 1.28 $\pm$ 0.16 times as strong as the expected linear response.
Similar to the two layer neural network, this model treats pyramidal neurons as cells composed of two layers. The first layer is composed of many individual subunits (the dendrites), in which a set of terms is calculated based on the input vector. These terms are then summed up in the second layer, giving the cells overall sub threshold activity level.
In addition, an output nonlinearity $g$ may be applied to $a(x)$, giving $y = g(a)$. While this model is more realistic than a point neuron, it only characterizes subthreshold activity, and is therefore not useful when analyzing spiking behaviour.
This model is nearly identical to the two layer model, however now the Apical Tuft is being taken into consideration. The Apical Tuft serves as a third layer of computation which calculates a gain factor that is transmitted to the soma. This gain factor is simply a multiplier to the somatic output calculated in the Two Layer Model. At the Apical Tuft Itself, dendritic branches calculate a sigmoid function of their inputs, acting much like the typical dendritic subunits of the Two Layer Model. These responses are then summed at an integrating center, and converted into a gain factor, which is then transmitted to the soma.
This model came about from studies showing that the largest postsynaptic response in a pyramidal cell occured when activated synapses were all located within clusters of an intermediate size. In the model, these clusters are treated as neuron-like subunits called clusterons. Each synapse has a region of distance $D$ centered over it. If two synapses are activated, and the distance between them is less than $D/2$, then they are considered to be in the same subunit, and a multiplicative interaction occurs between the two of them. In particular, consider an input $x_j$ with a region of $D_j$ surrounding it.
where $w_j$ is the weight of the synapse.
A major drawback to this model is that it only considers excitatory inputs, and does not take into consideration spatial characteristics of the branching dendritic tree.
Coincidence detection is believed to be important in the induction of synaptic plasticity and long term potentiation (LTP). Following a somatic action potential, a Back Propagating Action Potential (BPAP) sends a wave of depolarization towards the distal dendrites of the cell. If the BPAP reaches the dendrite at the same time as an EPSP is induced (or slightly before), the depolarization caused by the BPAP and the EPSP becomes much greater than their expected linear sum. This depolarization is mediated by dendritic sodium channels which open when EPSP's bring the dendritic membrane potential into a range where a BPAP will cause a threshold to be crossed.This wave of depolarization can then travel back to the soma, and while it will not affect the initial action potential amplitude, it causes a large afterdepolarization which may induce the soma to fire another action potential (leading to burst firing). The time window between the arrival of a BPAP and the EPSP for coincidence detection to occur appears to be similar to the time window between pre and post synaptic neuron firing in the induction of LTP (the strengthening of a synapse when pre and postsynaptic neurons fire simultaneously). This implies that coincidence detection may be important in LTP, synaptic plasticity, and long term memory.
EPSP must occur simultaneously with, or less than 10 - 15 ms before somatic action potential for coincidence detection to occur.
Maximal amplification of BPAP occurs when EPSP occurs less than 3 ms before a somatic action potential.
BPAP amplification was only observed at dendritic distances greater than 450 μm.
Gasparini, S., Migliore, M., and Magee., J.C. (2004). On the Initiation and Propagation of Dendritic Spikes in CA1 Pyramidal Neurons. The Journal of Neuroscience, 24(49):11046-11056.
Megias, M., Emri, ZS., Freund, T.F., and Guly, A. i. (2001) Total number and distribution of inhibitory and excitatory synapses on hippocampal CA1 pyramidal cells. Neuroscience, 102, 3:527-540.
Poirazi, P., Brannon, T., and Mel, B.W. (2003). Pyramidal Neuron as Two-Layer Neural Network. Neuron, 37, 989-999.
Polsky, A., Mel, B.W., and Schiller, J. (2004). Computational subunits in thin dendrites of pyramidal cells. Nature neuroscience, 7, 6:621-627.
Spruston, N. (2008). Pyramidal neurons: dendritic structure and synaptic integration. Nature neuroscience reviews. 9:206-221. | CommonCrawl |
Семинар Теорија вероватноћа и математичка статистика, 9, 10. и 11. јул 2018.
Апстракт: We investigate the randomized Karlin model with parameter $\beta\in(0,1)$, which is based on an infinite urn scheme. It has been shown before that when the randomization is bounded, the so-called odd-occupancy process scales to a fractional Brownian motion with Hurst index $\beta/2\in(0,1/2)$. We show that when the randomization is heavy-tailed with index $\alpha\in(0,2)$, then the odd-occupancy process scales to a new $(\beta/\alpha)$-self-similar symmetric $\alpha$-stable process with stationary increments.
Апстракт: We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one regime, our results exhibit limits that are not among the classical extreme value distributions. Restricted to the one-dimensional case, the distributions we obtain interpolate, in the appropriate parameter range, the alpha-Frechet distribution and the skewed \alpha-stable distribution. In general, the limit is a new family of stationary and self-similar random sup-measures with parameters alpha in (0,infty) and beta in (0,1), with representations based on intersections of independent beta-stable regenerative sets. The tail of the limit random sup-measure on each interval with finite positive length is regularly varying with index -alpha. The intriguing structure of these random sup-measures is due to intersections of independent beta-stable regenerative sets and the fact that the number of such sets intersecting simultaneously increases to infinity as beta increases to one.
Наслов предавања: EXTREME VALUE ANALYSIS WITHOUT THE LARGEST VALUES: WHAT CAN BE DONE?
Апстракт: Motivated by an analysis of the degree distributions in a large social network, we are concerned with the analysis of heavy-tailed data when a portion of the extreme values are unavailable. We focus on the Hill estimator, which plays a starring role in heavy-tailed modeling. The Hill estimator for this data exhibited a smooth and increasing "sample path" as a function of the number of upper order statistics used in constructing the estimator. This behavior became more apparent as we artificially removed more of the upper order statistics. Building on this observation, we introduce a new parameterization into the Hill estimator that is a function of ? and ?, that correspond, respectively, to the proportion of extreme values that are unavailable and the proportion of upper order statistics used normalized Hill estimator to a Gaussian random field. An estimation procedure is developed based on the limit theory to estimate the number of missing extremes and extreme value parameters including the tail index and the bias of Hill's estimate. We illustrate how this approach works in both simulations and real data examples. | CommonCrawl |
This article describes the extension of recent methods for a posteriori error estimation such as dual-weighted residual methods to node-centered finite volume discretizations of second order elliptic boundary value problems including upwind discretizations. It is shown how different sources of errors, in particular modeling errors and discretization errors, can be estimated with respect to a user-defined output functional.
We prove the $L^\infty(L^\infty)$-boundedness of a higher-order shock-capturing streamline-diffusion DG-method based on polynomials of degree $p\geq 0$ for general scalar conservation laws. The estimate is given for the case of several space dimensions and for conservation laws with initial and boundary conditions.
The paper presents results on piecewise polynomial approximations of tensor product type in Sobolev-Slobodecki spaces by various interpolation and projection techniques, on error estimates for quadrature rules and projection operators based on hierarchical bases, and on inverse inequalities. The main focus is directed to applications to discrete conservation laws.
Vol 6: Azacitidine in patients with WHO-defined AML - Results of 155 patients from the Austrian Azacitidine Registry of the AGMT-Study Group.
Vol 12: Position paper on the importance of psychosocial factors in cardiology: Update 2013. | CommonCrawl |
In his text Unreasonably Big Physics, Tetragraviton classifies the Texan SSC collider as marginally reasonable but other proposed projects are said to be unreasonable.
They include a wonderful 2017 collider proposal in the Gulf of Mexico. The structure would host some new, potentially clever 4-tesla dipoles and would be located 100 meters under the sea level between Houston and Merida.
The collision energy would be intriguing \(2\times 250\TeV=500\TeV\), almost 40 times higher than the current LHC beam, and the luminosity would trump the LHC by orders of magnitude, too. The depth is high enough not to annoy fish and to protect the tunnel against the hurricanes and the radius-of-300-kilometers ring would be far enough from beaches not to interfere with shipping. Quite generally, I think that the potentially brilliant idea that sea colliders could be more practical than the underground colliders should be honestly considered.
The cost is supposed to be comparable to the planned Chinese or European colliders – which means it's supposed to be very cheap. The adjective "cheap" is mine and unavoidably involves some subjective judgement. But I simply think that if someone finds a collider of this energy and this price "expensive", then he dislikes particle physics and it's bad if Tetragraviton belongs to that set.
I don't know exactly why you would want to do that but I know why we want a \(500\TeV\) collider. Every child knows why we want a \(500\TeV\) collider (or a plastic pony for Missy).
Well, I completely disagree with Tetragraviton that the Gulf of Mexico collider is unreasonable or impossible. If the calculations are right, it's actually a proposal you can't refuse. For the funds that only exceed the cost of the LHC by a small factor, we could increase the energy by a factor of 40. Isn't it wonderful?
He's not terribly specific about the arguments for his criticism but in between the lines, it seems that he finds tens of billions of dollars to be too much. Those amounts may be higher than his wealth but he's not supposed to pay for the whole thing. The world's GDP approaches $100 trillion a year. It's around $250 billion a day – including weekends – or $10 billion per hour.
Every hour, the world produces the wealth equal to the cost of the LHC dollider so the Gulf of Mexico collider could be equivalent just to few hours of the mankind's productive activity. Of course, some people may claim that it's arrogant to assume that the whole mankind contributes to something as esoteric as particle physics.
First of all, it's not arrogant – on the contrary, it's arrogant for someone to suggest that a human being could ignore particle physics. Take into the account that the extraterrestrials are watching us: Wouldn't you be terribly ashamed of the human race if it acts as a bunch of stinky pigs who won't dedicate even a few hours of their work to such groundbreaking projects of the global importance? Second of all, even if you compare the tens of billions of dollars to the funding for science only, it's small. Science may be getting roughly 1% of the global GDP which is one trillion dollars per year (globally). So such a unique project could still be equivalent just to weeks of the global spending for science.
It's totally counterproductive for Tetragraviton to spread his small-aß sentiments indicating that science shouldn't deserve tens-of-billions-of-dollars scientific projects. The mankind is getting richer, the rich enough countries can surely feed everybody and the poor countries may join as well, and there will be an increasing pile of excess cash (and workers who want some well-defined job).
It's natural for creative people and especially dreamers to have increasingly demanding visions and unless we screw something a big time, it should be increasingly easy to make these dreams come true. On top of that, every investment should compare costs and benefits – their differences and ratios. If a collider project increases the center-of-mass energy much more significantly than costs, then it simply deserves the particle physicists', engineers', and sponsors' attention.
Pure science will probably not get above $100 billion projects soon. But if you had some big project that would be somewhat scientific but also apparently very useful for lots of people or nations, I do believe that even multi-trillion projects should be possible.
The whole Apollo Program (whose outcome were all the men on the Moon) cost $25 billion of 1973 dollars which is translated to $110 billion of 2018 dollars. NASA's spending as a percentage of the U.S. Fed government's expenses peaked in 1966, under Lyndon Johnson's watch, when it was 4.41% or $6 (old big) billion. That one-year spending for one "applied scientific" institution already trumps the cost of the LHC when you convert it to current dollars.
Lunar missions have become boring for the taxpayers but other things may get hot again. Maybe there are great reasons to drill a hole through the Earth, build a tunnel around the Earth's circumference, or bring the ocean to the middle of Sahara, among hundreds of similar things I could generate effectively. Tetragraviton represents a textbook example of what Czechs call a near-wall-šitter (přizdisráč), a frightened man without self-confidence and ambitions. The Academia is full of this attitude, especially if you look at some typical bureaucrats in the scientific environment (who got to their chair mostly for their invisibility). But that's not the right attitude for those who should make similar big decisions. That's not the attitude of the men who change the world. That's not the men whom I really admire.
Does neutron decay to dark matter? | CommonCrawl |
I do not know any multimodal distributions.
Why are all known distributions unimodal? Is there any "famous" distribution that have more than one mode?
Of course, mixtures of distributions are often multimodal, but I would like to know whether there exist any "non-mixture" distributions that have more than one mode.
The first part of the question is answered in comments to the question: plenty of "brand-name" distributions are multimodal, such as any Beta$(a,b)$ distribution with $a\lt 1$ and $b\lt 1$. Let's turn, then, to the second part of the question.
All discrete distributions are clearly mixtures (of atoms, which are unimodal).
I will show that most continuous distributions are also mixtures of unimodal distributions. The intuition behind this is simple: we can "sand off" bumps from a bumpy graph of a PDF, one by one, until the graph is horizontal. The bumps become the mixture components, each of which is obviously unimodal.
Consequently, except perhaps for some unusual distributions whose PDFs are highly discontinuous, the answer to the question is "none": all multimodal distributions that are absolutely continuous, discrete, or a combination of those two are mixtures of unimodal distributions.
$f$ has a constant value on $m,$ say $y$.
$f$ is not constant on any interval that strictly contains $m$.
There exists a positive number $\epsilon$ such that the maximum value of $f$ attained on $[x_l-\epsilon, x_u+\epsilon]$ equals $y$.
Let $m = [x_l, x_u]$ be any mode of $f$. Because $f$ is continuous, there are intervals $[x_l^\prime, x_u^\prime]$ containing $m$ for which $f$ is nondecreasing in $[x_l^\prime, x_l]$ (which is a proper interval, not just a point) and nonincreasing in $[x_u, x_u^\prime]$ (which is also a proper interval). Let $x_l^\prime$ be the infinimum of all such values and $x_u^\prime$ the supremum of all such values.
This construction has defined one "hump" on the graph of $f$ extending from $x_l^\prime$ to $x_u^\prime$. Let $y$ be the larger of $f(x_l^\prime)$ and $f(x_u^\prime)$. By construction, the set of points $x$ in $[x_l^\prime, x_u^\prime]$ for which $f(x)\ge y$ is a proper interval $m^\prime$ strictly containing $m$ (because it contains either the whole of $[x_l^\prime, x_l]$ or $[x_u, x_u^\prime]$).
In this illustration of a multimodal PDF, a mode $m=[0,0]$ is identified by a red dot on the horizontal axis. The horizontal extent of the red portion of the fill is the interval $m^\prime$: it is the base of the hump determined by the mode $m$. The base of that hump is at height $y\approx 0.16$. The original PDF is the sum of the red fill and the blue fill. Notice that the blue fill only has one mode near $2$; the original mode at $[0,0]$ has been removed.
when $x \in m^\prime$ and $f_m(x)=0$ otherwise. (This makes $f_m$ a continuous function, incidentally.) The numerator is the amount by which $f$ rises above $y$ and the denominator $p_m$ is the area between the graph of $f$ and $y$. Thus $f_m$ is non-negative and has total area $1$: it is the PDF of a probability distribution. By construction it has a unique mode $m$.
is a mixture of the unimodal PDF $f_m$ and the PDF $f_m^\prime$.
... plus or minus some skewness or some discontinuities? When the question is posed thus, the Beta distribution would not be a valid counter example.
It appears the OP's conjecture has some validity: most common brand name distributions do not allow for more than one interior mode. There may be theoretical reasons for this. For example, any distribution that is a member of the Pearson family (which includes the Beta) will necessarily be (interior) unimodal, as a consequence of the parent differential eqn that defines the entire family. And the Pearson family nests most of the best-known brand names.
That you mightn't think of any doesn't mean there aren't any.
I can name "known" distributions that aren't unimodal.
For example, a Beta distribution with $\alpha$ and $\beta$ both $<1$.
Mixture distributions are certainly known, and many of those are multimodal.
where $\varphi$ is the PDF of a standard Gaussian.
Not the answer you're looking for? Browse other questions tagged distributions mode or ask your own question.
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