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# Genetics

Genetics is the science of heredity. It includes the study of genes and the inheritance of variation and traits of living organisms. It is an important part of biology and gives the basic rules on which evolution acts.

The fact that living things inherit traits from their parents is known for a long time. The modern science of genetics seeks to understand the process of inheritance.

Some of the human diseases are related to genetics. The unit of genetic structure is deoxyribonucleic acid (DNA). DNA is organized in the form of genes. Each gene is responsible for one or a set of functions. Some human diseases are related to a single gene. Some other diseases of genetic origin are related to more than one gene. In a group of persons having the same genetic disease, a similar pattern of genes is observed. In such cases, they may respond to a drug in a similar fashion. This observation has led to the development of personalized medicine. The study of genetics is important for personalized medicine. The principles of personalized medicines can help in reducing adverse reactions to a set of persons with genetic similarity

Genetics is the study of genes. It involves knowledge about genes and their functioning. Genes are located inside the nucleus and their sequence carries information. This information determines how living organisms inherit various features which are called phenotypic traits. Genetics identifies which features are inherited and explains how these features pass from one generation to another. In addition to inheritance, it studies how genes are turned on and off to control which substances are made in a cell-gene expression; and how a cell divides mitosis and meiosis.

Some phenotypic traits like the color of the eye are clearly visible. Other phenotypic traits like variation in an enzyme may not be easily visible. Traits determined by genes can be modified by environment e.g. height of a person is determined both by genetics and nutrition.

## Chromosomes

A chromosome is a DNA molecule with part or all of the genetic materials (genome) of an organism. Chromosomes are normally visible under a light microscope only when the cell is undergoing the metaphase of cell division. Before cell division, every chromosome is copied once during the S phase. The copy is joined to the original by a centromere resulting in an X-shaped structure. (Fig. 1.1).

The original chromosome and the copy are now called sister chromatids, During metaphase, when a chromosome is in its most condensed state, the X-shaped structure is called a metaphase chromosome. In this highly condensed form chromosomes are identified under the light microscope.

Compaction of the duplicated chromosomes during cell division (mitosis/meiosis) results either in a four-arm structure (see figure). The shape of a chromosome depends on the location of the centromere. If the chromatin is located in the center the two arms are equal. If the centromere is located near one of the arms the structure is asymmetrical as shown in Fig. 1.1. Chromosomal recombination during meiosis and subsequent sexual reproduction plays a significant role in genetic diversity. If these structures are manipulated incorrectly, the cell may undergo mitotic catastrophe and die; or it may unexpectedly evade apoptosis leading to the progression of cancer.

Nuclear chromosomes are packed by proteins into a condensed structure called chromatin. This allows very long DNA molecules to fit into the tiny cell nucleus. The structure of chromosomes and chromatin varies through the cell cycle. They are even more condensed than chromatin and are an essential unit for cellular division. They must be replicated, divided, and passed successfully to their daughter cells to ensure that the genetic diversity and survival of the progeny are maintained. They may exist as either duplicated or unduplicated. Unduplicated chromosomes are single double helixes, while duplicated chromosomes contain two identical copies, called sister chromatids, and are joined by a centromere.

In the early stages of mitosis or meiosis, the chromatin double helix becomes more and more condensed. During this stage transcription stops and genetic material becomes a compact transportable form. The compact form makes individual chromosomes visible. Mitotic metaphase chromosomes are best described by a linearly organized, longitudinally compressed array of consecutive chromatin loops.

During mitosis, microtubules grow from centrosomes located at opposite ends of the cell and attach to the centromere at specialized structures called kinetochores, one of which is present on each sister chromatid. A special DNA base sequence in the region of kinetochores provides longer standing attachment in this region. The microtubules then pull the chromatids apart towards the centrosomes so that each daughter cell inherits one set of chromatids. Once the cells have divided, the chromatids are uncoiled and DNA can again be transcribed.

### Human Chromosomes

Chromosomes in human beings can be divided into two types: autosomes and allosomes. The autosomes contain genetic hereditary information related to body characteristics except for sex. Allosomes are sex chromosomes. Certain genetic traits are linked to a person’s sex and are passed through the sex chromosome. All chromosomes act in the same way during cell division. Human cells have 22 pairs of autosomes and one pair of allosomes. The total makes 23 pairs or 46 chromosomes. The following table gives the number of genes and % of the basis for each gene.

It is to be noted that for the same number of genes, total base pairs may vary because some of the sequences are silent. Hence, figures in column number 2 do not match exactly with the % basis in column number 3.

## GENES AND DNA

### DNA

The molecular basis for genetics is deoxyribonucleic acid (DNA). It consists of a chain made from four types of nucleotide subunits, each composed of a five-carbon sugar, called 2-deoxyribose; a phosphate group, and one of four bases: Adenine (A), Cytosine (C). Guanine (G) and Thymine (T).

Two chains of DNA twist around each other to form a DNA double helix with the phosphate-sugar backbone spiraling around the outside and the bases pointing inwards. In the inner part of the helix adenine base pairs with thymine while guanine base pairs with cytosine. The specificity of base pairing occurs because A pairs with T with two hydrogen bonds while C pairs with G with three hydrogen bonds. Thus, two strands in a double helix are complementary, with their sequence of bases matching such that As of one strand pair with Ts of complementary strand and Cs of one strand pair with Gs of complementary strand.

Due to the chemical composition of the pentose residues of the bases, DNA strands have directionality. One end of a DNA polymer contains an exposed hydroxyl group on the deoxyribose; this is known as the 3′ end of the molecule. The other end contains an exposed phosphate group; this is the 5′ end. The two strands of a double helix run in opposite directions. Nucleic acid synthesis, including DNA replication and transcription, occurs in the 5′ →→ 3′ direction because new nucleotides are added via a dehydration reaction that uses the exposed 3′ hydroxyl as a nucleophile.

The expression of genes encoded in DNA begins by transcribing the gene into RNA, the second type of nucleic acid which is similar to DNA with the only difference that sugar content in RNA is ribose, unlike deoxyribose in DNA. RNA also contains the base uracil in place of thymine. RNA molecules are less stable than DNA and are typically single-stranded. There are two types of RNA: t RNA and m RNA. t RNA is a transfer RNA that has a three-nucleotide sequence called codon at its one end. At another end, t RNA carries one amino acid. Every amino acid has a different t RNA for the purpose of translocation.

### Gene

A gene is a locus or region of DNA which is made up of nucleotides and is the molecular unit of heredity. The transmission of genes to an organism’s offspring is the basis of the inheritance of phenotypic traits. These genes make up different DNA sequences called genotypes. Genotypes along with environmental and developmental factors determine what the phenotypes will be. A set of genes together is called polygenes. Many biological traits are under the influence of polygenes and gene-environmental interactions. Some genetic traits like the color of the eye are instantly visible. Some other traits like blood groups may not be instantly visible.

Genes can acquire mutations in their sequence, leading to different variants known as alleles. These alleles encode slightly different versions of a protein, which cause different phenotypical traits. Usage of the term “having a gene” typically refers to containing a different allele of the same gene. Genes evolve due to natural selection or struggle for existence leading to the survival of the fittest amongst alleles.

The concept of a gene continues to be refined as new phenomena are discovered. Some viruses store their genome in RNA instead of DNA and some gene products are functional non-coding RNAs. Therefore a broad, modern working definition of a gene is any discrete locus of heritable, genomic sequence which affects an organism’s traits by being expressed as a functional product or by regulation of gene expression.

#### Structure of a Gene

The structure of a gene consists of many elements of which the actual protein coding sequence is often only a small part. These include DNA regions that are not transcribed as well as untranslated regions of the RNA.

All genes contain a regulatory sequence that is required for their expression. In order to be expressed, genes require a promoter sequence. The promoter is recognized and bound by transcription factors and RNA polymerase to initiate transcription. A gene can have more than one promoter resulting in m RNAs which differ in how far they extend in 5′ end. Promoter regions have a consensus sequence, however, highly transcribed genes have strong promoter sequences; while others have weak promoters which bind poorly and initiate transcription less frequently.

In addition, genes can have regulatory regions upstream or downstream to the open reading frame. These act by binding to transcription factors which then cause the DNA to loop so that the regulatory sequence becomes close to the RNA polymerase binding site. There are enhancer or silencer regions in a gene. Enhancers increase transcription by binding an activator protein which then helps to recruit the RNA polymerase to the promoter. Conversely, silencers bind repressor proteins and make the DNA less available for RNA polymerase.

The transcribed pre-cum RNA contains untranslated regions at both ends which contain a ribosome binding site, terminator and start and stop codons. In addition, open reading frames contain untranslated introns which are removed before exons are translated. The sequences at the end of introns dictate the splice sites to generate the final mature m RNA which encodes the protein or RNA product.

Many genes are organized into operons, with multiple coding sequences which are transcribed as a unit. The genes in an operon are transcribed as a continuous m RNA: referred to as polycistronic m RNA. The term cistron is equivalent to a gene. The transcription of an operon’s m RNA is often controlled by a repressor which can occur in an active or inactive state depending on the presence of certain metabolites. When active, the repressor binds to a DNA sequence at the beginning of the operon, called the operator region, and represses transcription of the operon. When the repressor is inactive, the transcription of the operon can occur. The products of operon genes typically have related functions and are involved in the same regulatory network.

### Functions of Genes:

Gene Expression:

In all organisms, two steps are required to read the information encoded in a gene’s DNA and produce the protein which it specifies

• First, the gene’s DNA is transcribed to m RNA.
• Second, that m RNA is translated to protein.

RNA coding genes must still go through the first step but are not translated into protein. The process of producing a biologically functional molecule of either RNA or protein is called gene expression, and the resulting molecule is called a gene product.

Genetic Code:

The nucleotide sequence of a gene’s DNA specifies the amino acid sequence of a protein through the genetic code. Sets of three nucleotides, called codons, each corresponds to a specific amino acid.

In addition, a start codon and three stop codons indicate the beginning and end of the protein-coding region. For four nucleotides there are 4³= 64 possible codons. There are 20 standard amino acids. Thus, multiple codons can specify the same amino acid. The correspondence between codons and amino acids is universal amongst all known living organisms.

#### Transcription:

Transcription produces a single-stranded m RNA, whose nucleotide sequence is complementary to the DNA from which it is transcribed. The m RNA acts as an intermediate between a gene and the final protein. The transcription is performed by an enzyme called RNA polymerase, which reads the template strands in the 3’5′ direction and synthesizes the RNA from 5’3′. To initiate transcription, the polymerase first recognizes and binds to the promoter region of the gene. Thus, a major mechanism of gene regulation is blocking or sequestering the promoter region, either by tight binding to repressor genes or by organizing the DNA so that the promoter region is not accessible.

#### Translation:

The translation is the process by which a mature RNA molecule is used as a template for synthesizing a new protein. The process is carried out by ribosomes, large complexes of RNA and protein responsible for carrying out chemical reactions to add new amino acids to a growing polypeptide chain by the formation of peptide bonds. The genetic code is read as three nucleotides at a time, in units called codons via interactions with t RNA Each t RNA has three unpaired bases which are known as anticodon and are complementary to the codon on m RNA. The t RNA is covalently attached to the amino acid specified by the complementary codon. When the t RNA binds to its complementary codon in an m RNA strand, the ribosome attaches its amino acid cargo to the new polypeptide chain, synthesized from amino-terminus to carboxyl-terminal. During and after synthesis, most new proteins must hold to their active three-dimensional structure before they can carry out their cellular functions.

#### Regulation:

Genes are regulated so that they are expressed only when the protein is needed. because expression draws on limited resources. A cell regulates its gene expression based on the external environment, internal environment, and its specific role in a multicellular organism. The external environment includes available nutrients, temperature, and other stresses. The internal environment includes the cell division cycle, metabolism, infection status, etc. A specific role includes whether the cell is part of the bone, skeletal muscle, or smooth muscle, etc. gene expression can be regulated at any step: from transcriptional initiation to RNA processing or to post-translational modification of the protein.

#### RNA Genes:

A typical protein-coding gene is first copied into RNA as an intermediate during the manufacture of a protein. In other cases, the RNA molecules are the actual functional products, as in the synthesis of ribosomal RNA and t RNA. Some RNAs known as ribozymes are capable of enzymatic function, and micro RNA has a regulatory role. The DNA sequences from which such RNAs are transcribed are called non-coding RNA genes.

Some viruses store their entire genomes in the form of RNA and do not contain DNA at all.

#### Genome:

The genome is the total genetic material of an organism and includes both genes and non-coding sequences.

The size of the genome and the number of genes it encodes varies widely between organisms Number of base pairs of DNA in the human genome is known since the 1960s Sequencing of genes in the human genome project indicated that the total number of protein-coding genes is only about 20,000 with 13 genes encoded on the mitochondrial genome. Of the human genome, only 1-2% consist of protein-coding genes, with the remainder being non-coding DNA such as introns, retrotransposons, and non-coding RNAs.

## Pattern of Inheritance

Organisms inherit their genes from their parents. There are two copies of each chromosome because they inherit one complete set from each parent.

### Mendelian Inheritance:

It involves two terms: phenotype and genotype. Phenotypes are observable physical and/or behavioral characteristics e.g. color of the eye. Genotypes are a particular set of genes. Each gene specifies a particular trait with a different sequence of the gene, called alleles, giving rise to different phenotypes. Thus, one genotype can lead to more phenotypes, probably because of environmental alterations.

Alleles at a locus may be dominant or recessive. Dominant alleles give rise to their corresponding phenotypes when paired with any other allele for the same trait. In the case of recessive alleles, they give rise to the corresponding phenotype only when paired with another copy of the same allele. If you know the genotypes of the organisms, you can determine which alleles are dominant and which are recessive. Mendel’s work demonstrated that alleles assort independently in the production of gametes or germ cells, ensuring variation in the next generation. Mendelian inheritance continues to be a good model for many traits determined by single genes. Genetic disorders linked to a single gene follow Mendelian inheritance. It does not include the physical process of DNA replication and cell division. Molecular details of genetics were not known during Mendelian inheritance.

### Molecular inheritance:

The duplication and transmission of genetic material from one generation of cells to the next is the basis of molecular inheritance. It is the link between classical and molecular genetics. Organisms inherit the characteristics of their parents because the cells of the offspring contain copies of the genes of their parent’s cells. During cell division the first. the stage is meiosis, wherein the cells get one copy of each gene. The gametes produced by the female are eggs/ova and those produced by males are sperms. Two gametes fuse to form a diploid fertilized egg. a single cell having two sets of genes, one each from the mother and the father.

During meiosis, an event called genetics recombination or crossing over occurs, in which a length of DNA on one chromatid is swapped with a length of DNA on the corresponding homologous non-sister chromatid. This can result in the re-assortment of otherwise linked alleles. The Mendelian principle of independent assortment asserts that each of a parent’s two genes for each trait will sort independently into gametes; which allele an organism inherits for one trait is unrelated to which allele it inherits for another trait. This is true for genes that do not reside on the same chromosome or are located very far from one another on the same chromosome. The closer two genes lie on the same chromosome, the more closely they will be associated with gametes and the more often they will appear together. Genes that are very close are never separated because it is extremely unlikely that a cross-over point will occur between them. This is known as genetics linkage.

### Mutation:

DNA replication is extremely accurate; however, errors can occur. The errors are called mutations. Small mutations can be caused by DNA replication. It can take different forms In point mutations a single base is altered. In frameshift mutations, a single base is inserted or deleted. Either of these mutations can change the gene either as a missense or nonsense. In the case of missense, there is a change in codon to encode a different amino acid. In the case of nonsense, there is a premature stop codon leading to incomplete transcription/translation. Large mutations can be caused by errors in recombination to cause chromosomal abnormalities such as duplication, deletion, rearrangement, or inversion of large sections of a chromosome. In addition, DNA repair mechanisms can introduce mutational errors while repairing physical damage to the molecule. The repair, even with mutation, is more important to survival than restoring an exact copy, e.g. repairing of double-strand breaks.

When multiple different alleles for a gene are present in a species’ population, it is called polymorphic. Most different alleles are functionally equivalent, however, some alleles can give rise to different phenotypes. A gene’s most common allele is termed as wild type, and rare alleles are called mutants. Genetics variation in the relative frequency of different alleles in a population is due to both natural selection as well as genetic drift. The wild types are not necessarily the ancestors of less common alleles, nor are they fitter than the modified alleles.

Mutations can be of different types. Those having no effect on an organism’s phenotype are called silent mutations. Some mutations do not change the amino acid sequence because multiple codons encode the same amino acid. They are called synonymous mutations. Some mutations can be silent, but the changed amino acid sequence of the protein still preserves the functions of the protein. They are called conservative mutations. However many mutations are deleterious or even lethal; such mutations are removed from populations by natural selection. Genetic disorders are the result of deleterious mutations and can be due to spontaneous mutation in the affected individual, or can be inherited. Over and above this, a small fraction of mutations is beneficial, leading to better fitness, and are extremely important for evolution.

Sequence Homology

Genes with a most recent common ancestor, and thus a shared evolutionary ancestry are called homologs. These genes can appear from gene duplication within an organism’s genome, where they are known as paralogous genes. Alternatively, they are the result of diversions of the genes after a speciation event where they are known as orthologous genes; such genes often perform the same or similar functions in the related organism.

The relationship between genes can be measured by comparing the sequence alignment of their DNA. The degree of sequence similarity between homologous genes is called the conserved sequence.

#### Origin of New Genes

The most common source of new genes is gene duplication, which creates copy number variation of an existing gene in the genome. The resulting genes may then diverge in sequence and in function. Sets of genes formed in this manner are called gene families. Gene duplications and losses within a family are common and represent a major source of biological diversity. Sometimes gene duplication may result in a non-functional copy of a gene, or a functional copy may have mutations leading to loss of function. Such non-functional genes are called pseudogenes.

Some sequences show no similarity to existing genes and are less common than gene duplicates. They are called orphan genes. Horizontal gene transfer refers to the transfer of genetic material through a mechanism other than reproduction. This mechanism is a common source of new genes. Sometimes they contribute more to genetic variation than gene duplication. It is a common means of spreading antibiotic resistance, virulence, and adaptive metabolic functions.

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• ### Complete event-by-event $\alpha$/$\gamma(\beta)$ separation in a full-size TeO$_2$ CUORE bolometer by Neganov-Luke-magnified light detection(1710.03459)

April 25, 2018 hep-ex, physics.ins-det
In the present work, we describe the results obtained with a large ($\approx 133$ cm$^3$) TeO$_2$ bolometer, with a view to a search for neutrinoless double-beta decay ($0\nu\beta\beta$) of $^{130}$Te. We demonstrate an efficient $\alpha$ particle discrimination (99.9\%) with a high acceptance of the $0\nu\beta\beta$ signal (about 96\%), expected at $\approx 2.5$ MeV. This unprecedented result was possible thanks to the superior performance (10 eV rms baseline noise) of a Neganov-Luke-assisted germanium bolometer used to detect a tiny (70 eV) light signal from the TeO$_2$ detector, dominated by $\gamma$($\beta$)-induced Cherenkov radiation but exhibiting also a clear scintillation component. The obtained results represent a major breakthrough towards the TeO$_2$-based version of CUORE Upgrade with Particle IDentification (CUPID), a ton-scale cryogenic $0\nu\beta\beta$ experiment proposed as a follow-up to the CUORE project with particle identification. The CUORE experiment began recently a search for neutrinoless double-beta decay of $^{130}$Te with an array of 988 125-cm$^3$ TeO$_2$ bolometers. The lack of $\alpha$ discrimination in CUORE makes $\alpha$ decays at the detector surface the dominant background component, at the level of $\approx 0.01$ counts/(keV kg y) in the region of interest. We show here, for the first time with a CUORE-size bolometer and using the same technology as CUORE for the readout of both heat and light signals, that surface $\alpha$ background can be fully rejected.
• ### Terahertz emission from laser-driven gas-plasmas: a plasmonic point of view(1803.06889)

We investigate terahertz generation in two-color laser-induced gas plasmas. Resonances attributed to plasmonic effects are shown to broaden the emission spectra significantly, when electrons are excited along a direction where the plasma size is smaller than the plasma wavelength. We demonstrate that such resonances can be turned on and off by changing the polarization properties of elliptically-shaped driving laser-pulses. Both 3D Maxwell consistent simulations and experimental results confirm that a significant THz pulse shortening and spectral broadening can be achieved by exploiting the transverse driving laser-beam shape as an additional degree of freedom.
• ### QUBIC - The Q&U Bolometric Interferometer for Cosmology - A novel way to look at the polarized Cosmic Microwave Background(1801.03730)

Jan. 11, 2018 astro-ph.IM
In this paper we describe QUBIC, an experiment that takes up the challenge posed by the detection of primordial gravitational waves with a novel approach, that combines the sensitivity of state-of-the art bolometric detectors with the systematic effects control typical of interferometers. The so-called "self-calibration" is a technique deeply rooted in the interferometric nature of the instrument and allows us to clean the measured data from instrumental effects. The first module of QUBIC is a dual band instrument (150 GHz and 220 GHz) that will be deployed in Argentina during the Fall 2018.
• ### Environnement-insensitive calibration for cryogenic broadband microwave reflectometry measurements(1801.00436)

Jan. 1, 2018 cond-mat.supr-con
The characteristic frequencies of a system provide important information on the phenomena that govern its physical properties. In this framework, there has recently been renewed interest in cryogenic microwave characterization for condensed matter systems since it allows to probe energy scales of the order of a few $\mu$eV. However, broadband measurements of the absolute value of a sample response in this frequency range are extremely sensitive to its environment and require a careful calibration. In this paper, we present an original in situ calibration method for cryogenic broadband microwave reflectometry experiments that increases the sensitivity of such measurements. The calibration references are here provided by the sample itself, at three reference temperatures where its impedance is assumed or measured, and not by external standards as is usual. We compare the frequency-dependent complex impedance of an a-Nb$_{15}$Si$_{85}$ superconducting thin film obtained through this Sample-Based Calibration (SBC) and through a Open-Short-Load Standard Calibration (SC) when working at very low temperature (T < 1 K). While the SC only provides a qualitative evolution of the impedance across the superconducting transition, the SBC allows us to obtain the absolute response of the sample, which we compared with the relevant theories. The higher sensitivity of the SBC, supported both by the experimental results and by numerical calculations of the error propagation, is due to the definition of the calibration planes - as close as possible to the sample - which allows to completely get rid of the sample's electromagnetic environment.
• ### Development of $^{100}$Mo-containing scintillating bolometers for a high-sensitivity neutrinoless double-beta decay search(1704.01758)

Oct. 4, 2017 nucl-ex, physics.ins-det
This paper reports on the development of a technology involving $^{100}$Mo-enriched scintillating bolometers, compatible with the goals of CUPID, a proposed next-generation bolometric experiment to search for neutrinoless double-beta decay. Large mass ($\sim$1~kg), high optical quality, radiopure $^{100}$Mo-containing zinc and lithium molybdate crystals have been produced and used to develop high performance single detector modules based on 0.2--0.4~kg scintillating bolometers. In particular, the energy resolution of the lithium molybdate detectors near the $Q$-value of the double-beta transition of $^{100}$Mo (3034~keV) is 4--6~keV FWHM. The rejection of the $\alpha$-induced dominant background above 2.6~MeV is better than 8$\sigma$. Less than 10~$\mu$Bq/kg activity of $^{232}$Th ($^{228}$Th) and $^{226}$Ra in the crystals is ensured by boule recrystallization. The potential of $^{100}$Mo-enriched scintillating bolometers to perform high sensitivity double-beta decay searches has been demonstrated with only 10~kg$\times$d exposure: the two neutrino double-beta decay half-life of $^{100}$Mo has been measured with the up-to-date highest accuracy as $T_{1/2}$ = [6.90 $\pm$ 0.15(stat.) $\pm$ 0.37(syst.)] $\times$ 10$^{18}$~yr. Both crystallization and detector technologies favor lithium molybdate, which has been selected for the ongoing construction of the CUPID-0/Mo demonstrator, containing several kg of $^{100}$Mo.
• ### Sequential terahertz pulse generation by photoionization and coherent transition radiation in underdense relativistic plasmas(1709.04639)

Sept. 14, 2017 physics.plasm-ph
Terahertz (THz) emission by two-color, ultrashort optical pulses interacting with underdense helium gases at ultrahigh intensities ($> 10^{19}\,\mathrm{W/cm}^2$) is investigated by means of 3D particle-in-cell simulations. The THz field is shown to be produced by two mechanisms occurring sequentially, namely, photoionization-induced radiation (PIR) by the two-color pulse and coherent transition radiation (CTR) by the wakefield-accelerated electrons escaping the plasma. For plasmas of atomic densities $> 10^{17}\,\mathrm{cm}^{-3}$, CTR proves to be the dominant process, providing THz bursts with field strength as high as $100\,\mathrm{GV/m}$ and energy in excess of $1\,\mathrm{mJ}$. Analytical models are developed for both the PIR and CTR processes, which correctly reproduce the simulation data.
• ### Optimizing EDELWEISS detectors for low-mass WIMP searches(1707.04308)

July 12, 2017 physics.ins-det, astro-ph.CO
The physics potential of EDELWEISS detectors for the search of low-mass Weakly Interacting Massive Particles (WIMPs) is studied. Using a data-driven background model, projected exclusion limits are computed using frequentist and multivariate analysis approaches, namely profile likelihood and boosted decision tree. Both current and achievable experimental performance are considered. The optimal strategy for detector optimization depends critically on whether the emphasis is put on WIMP masses below or above $\sim$ 5 GeV/c$^2$. The projected sensitivity for the next phase of the EDELWEISS-III experiment at the Modane Underground Laboratory (LSM) for low-mass WIMP search is presented. By 2018 an upper limit on the spin-independent WIMP-nucleon cross-section of $\sigma_{SI} = 7 \times 10^{-42}$ cm$^2$ is expected for a WIMP mass in the range 2$-$5 GeV/c$^2$. The requirements for a future hundred-kilogram scale experiment designed to reach the bounds imposed by the coherent scattering of solar neutrinos are also described. By improving the ionization resolution down to 50 eV$_{ee}$, we show that such an experiment installed in an even lower background environment (e.g. at SNOLAB) should allow to observe about 80 $^8$B neutrino events after discrimination.
• ### Performance of the EDELWEISS-III experiment for direct dark matter searches(1706.01070)

We present the results of measurements demonstrating the efficiency of the EDELWEISS-III array of cryogenic germanium detectors for direct dark matter searches. The experimental setup and the FID (Fully Inter-Digitized) detector array is described, as well as the efficiency of the double measurement of heat and ionization signals in background rejection. For the whole set of 24 FID detectors used for coincidence studies, the baseline resolutions for the fiducial ionization energy are mainly below 0.7 keV$_{ee}$ (FHWM) whereas the baseline resolutions for heat energies are mainly below 1.5 keV$_{ee}$ (FWHM). The response to nuclear recoils as well as the very good discrimination capability of the FID design has been measured with an AmBe source. The surface $\beta$- and $\alpha$-decay rejection power of $R_{\rm surf} < 4 \times 10^{-5}$ per $\alpha$ at 90% C.L. has been determined with a $^{210}$Pb source, the rejection of bulk $\gamma$-ray events has been demonstrated using $\gamma$-calibrations with $^{133}$Ba sources leading to a value of $R_{\gamma{\rm -mis-fid}} < 2.5 \times 10^{-6}$ at 90% C.L.. The current levels of natural radioactivity measured in the detector array are shown as the rate of single $\gamma$ background. The fiducial volume fraction of the FID detectors has been measured to a weighted average value of $(74.6 \pm 0.4)\%$ using the cosmogenic activation of the $^{65}$Zn and $^{68,71}$Ge isotopes. The stability and uniformity of the detector response is also discussed. The achieved resolutions, thresholds and background levels of the upgraded EDELWEISS-III detectors in their setup are thus well suited to the direct search of WIMP dark matter over a large mass range.
• ### QUBIC Technical Design Report(1609.04372)

May 11, 2017 astro-ph.CO, astro-ph.IM
QUBIC is an instrument aiming at measuring the B mode polarisation anisotropies at medium scales angular scales (30-200 multipoles). The search for the primordial CMB B-mode polarization signal is challenging, because of many difficulties: smallness of the expected signal, instrumental systematics that could possibly induce polarization leakage from the large E signal into B, brighter than anticipated polarized foregrounds (dust) reducing to zero the initial hope of finding sky regions clean enough to have a direct primordial B-modes observation. The QUBIC instrument is designed to address all aspects of this challenge with a novel kind of instrument, a Bolometric Interferometer, combining the background-limited sensitivity of Transition-Edge-Sensors and the control of systematics allowed by the observation of interference fringe patterns, while operating at two frequencies to disentangle polarized foregrounds from primordial B mode polarization. Its characteristics are described in details in this Technological Design Report.
• ### Spectral dynamics of THz pulses generated by two-color laser filaments in air: The role of Kerr nonlinearities and pump wavelength(1611.05851)

Dec. 8, 2016 physics.optics
We theoretically and numerically study the influence of both instantaneous and Raman-delayed Kerr nonlinearities as well as a long-wavelength pump in the terahertz (THz) emissions produced by two-color femtosecond filaments in air. Although the Raman-delayed nonlinearity induced by air molecules weakens THz generation, four-wave mixing is found to impact the THz spectra accumulated upon propagation via self-, cross-phase modulations and self-steepening. Besides, using the local current theory, we show that the scaling of laser-to-THz conversion efficiency with the fundamental laser wavelength strongly depends on the relative phase between the two colors, the pulse duration and shape, rendering a universal scaling law impossible. Scaling laws in powers of the pump wavelength may only provide a rough estimate of the increase in the THz yield. We confront these results with comprehensive numerical simulations of strongly focused pulses and of filaments propagating over meter-range distances.
• ### Improved EDELWEISS-III sensitivity for low-mass WIMPs using a profile likelihood approach(1607.03367)

We report on a dark matter search for a Weakly Interacting Massive Particle (WIMP) in the mass range $m_\chi \in [4, 30]\,\mathrm{GeV}/c^2$ with the EDELWEISS-III experiment. A 2D profile likelihood analysis is performed on data from eight selected detectors with the lowest energy thresholds leading to a combined fiducial exposure of 496 kg-days. External backgrounds from $\gamma$- and $\beta$-radiation, recoils from $^{206}$Pb and neutrons as well as detector intrinsic backgrounds were modelled from data outside the region of interest and constrained in the analysis. The basic data selection and most of the background models are the same as those used in a previously published analysis based on Boosted Decision Trees (BDT). For the likelihood approach applied in the analysis presented here, a larger signal efficiency and a subtraction of the expected background lead to a higher sensitivity, especially for the lowest WIMP masses probed. No statistically significant signal was found and upper limits on the spin-independent WIMP-nucleon scattering cross section can be set with a hypothesis test based on the profile likelihood test statistics. The 90% C.L. exclusion limit set for WIMPs with $m_\chi = 4\,\mathrm{GeV/}c^2$ is $1.6 \times 10^{-39}\,\mathrm{cm^2}$, which is an improvement of a factor of seven with respect to the BDT-based analysis. For WIMP masses above $15\,\mathrm{GeV/}c^2$ the exclusion limits found with both analyses are in good agreement.
• ### Theory of terahertz emission from femtosecond-laser-induced micro-plasmas(1609.02734)

Sept. 9, 2016 physics.plasm-ph
We present a theoretical investigation of terahertz (THz) generation in laser-induced gas plasmas. The work is strongly motivated by recent experimental results on micro-plasmas, but our general findings are not limited to such a configuration. The electrons and ions are created by tunnel-ionization of neutral atoms, and the resulting plasma is heated by collisions. Electrons are driven by electromagnetic, convective and diffusive sources and produce a macroscopic current which is responsible for THz emission. The model naturally includes both, ionization current and transition-Cherenkov mechanisms for THz emission, which are usually investigated separately in the literature. The latter mechanism is shown to dominate for single-color multi-cycle lasers pulses, where the observed THz radiation originates from longitudinal electron currents. However, we find that the often discussed oscillations at the plasma frequency do not contribute to the THz emission spectrum. In order to predict the scaling of the conversion efficiency with pulse energy and focusing conditions, we propose a simplified description that is in excellent agreement with rigorous particle-in-cell simulations.
• ### Measurement of the cosmogenic activation of germanium detectors in EDELWEISS-III(1607.04560)

We present a measurement of the cosmogenic activation in the germanium cryogenic detectors of the EDELWEISS III direct dark matter search experiment. The decay rates measured in detectors with different exposures to cosmic rays above ground are converted into production rates of different isotopes. The measured production rates in units of nuclei/kg/day are 82 $\pm$ 21 for $^3$H, 2.8 $\pm$ 0.6 for $^{49}$V, 4.6 $\pm$ 0.7 for $^{55}$Fe, and 106 $\pm$ 13 for $^{65}$Zn. These results are the most accurate for these isotopes. A lower limit on the production rate of $^{68}$Ge of 74 nuclei/kg/day is also presented. They are compared to model predictions present in literature and to estimates calculated with the ACTIVIA code.
• ### Signals induced by charge-trapping in EDELWEISS FID detectors: analytical modeling and applications(1606.08097)

The EDELWEISS-III direct dark matter search experiment uses cryogenic HP-Ge detectors Fully covered with Inter-Digitized electrodes (FID). They are operated at low fields ($<1\;\mathrm{V/cm}$), and as a consequence charge-carrier trapping significantly affects both the ionization and heat energy measurements. This paper describes an analytical model of the signals induced by trapped charges in FID detectors based on the Shockley-Ramo theorem. It is used to demonstrate that veto electrodes, initially designed for the sole purpose of surface event rejection, can be used to provide a sensitivity to the depth of the energy deposits, characterize the trapping in the crystals, perform heat and ionization energy corrections and improve the ionization baseline resolutions. These procedures are applied successfully to actual data.
• ### Constraints on low-mass WIMPs from the EDELWEISS-III dark matter search(1603.05120)

We present the results of a search for elastic scattering from galactic dark matter in the form of Weakly Interacting Massive Particles (WIMPs) in the 4-30 GeV/$c^2$ mass range. We make use of a 582 kg-day fiducial exposure from an array of 800 g Germanium bolometers equipped with a set of interleaved electrodes with full surface coverage. We searched specifically for $\sim 2.5-20$ keV nuclear recoils inside the detector fiducial volume. As an illustration the number of observed events in the search for 5 (resp. 20) GeV/$c^2$ WIMPs are 9 (resp. 4), compared to an expected background of 6.1 (resp. 1.4). A 90% CL limit of $4.3\times 10^{-40}$ cm$^2$ (resp. $9.4\times 10^{-44}$ cm$^2$) is set on the spin-independent WIMP-nucleon scattering cross-section for 5 (resp. 20) GeV/$c^2$ WIMPs. This result represents a 41-fold improvement with respect to the previous EDELWEISS-II low-mass WIMP search for 7 GeV/$c^2$ WIMPs. The derived constraint is in tension with hints of WIMP signals from some recent experiments, thus confirming results obtained with different detection techniques.
• ### Picosecond laser filamentation in air(1601.02434)

The propagation of intense picosecond laser pulses in air in the presence of strong nonlinear self-action effects and air ionization is investigated experimentally and numerically. The model used for numerical analysis is based on the nonlinear propagator for the optical field coupled with the rate equations for the production of various ionic species and plasma temperature. Our results show that the phenomenon of plasma-driven intensity clamping, which is paramount in femtosecond laser filamentation, holds for picosecond pulses. Furthermore, the temporal pulse distortions are limited and the pulse fluence is also clamped. The resulting unique feature of the picosecond filamentation regime is the production of a broad, fully ionized air channel, continuous both longitudinally and transversely, which may be instrumental for numerous applications.
• ### Proceedings of the third French-Ukrainian workshop on the instrumentation developments for HEP(1512.07393)

Dec. 23, 2015 hep-ex, nucl-ex, physics.ins-det
The reports collected in these proceedings have been presented in the third French-Ukrainian workshop on the instrumentation developments for high-energy physics held at LAL, Orsay on October 15-16. The workshop was conducted in the scope of the IDEATE International Associated Laboratory (LIA). Joint developments between French and Ukrainian laboratories and universities as well as new proposals have been discussed. The main topics of the papers presented in the Proceedings are developments for accelerator and beam monitoring, detector developments, joint developments for large-scale high-energy and astroparticle physics projects, medical applications.
• ### Scintillating bolometers based on ZnMoO$_4$ and Zn$^{100}$MoO$_4$ crystals to search for 0$\nu$2$\beta$ decay of $^{100}$Mo (LUMINEU project): first tests at the Modane Underground Laboratory(1502.01161)

Feb. 4, 2015 nucl-ex, physics.ins-det
The technology of scintillating bolometers based on zinc molybdate (ZnMoO$_4$) crystals is under development within the LUMINEU project to search for 0$\nu$2$\beta$ decay of $^{100}$Mo with the goal to set the basis for large scale experiments capable to explore the inverted hierarchy region of the neutrino mass pattern. Advanced ZnMoO$_4$ crystal scintillators with mass of $\sim$~0.3 kg were developed and Zn$^{100}$MoO$_4$ crystal from enriched $^{100}$Mo was produced for the first time by using the low-thermal-gradient Czochralski technique. One ZnMoO$_4$ scintillator and two samples (59 g and 63 g) cut from the enriched boule were tested aboveground at milli-Kelvin temperature as scintillating bolometers showing a high detection performance. The first results of the low background measurements with three ZnMoO$_4$ and two enriched detectors installed in the EDELWEISS set-up at the Modane Underground Laboratory (France) are presented.
• ### Boosting terahertz generation in laser-field ionized gases using a sawtooth wave shape(1412.4995)

Broadband ultrashort terahertz (THz) pulses can be produced using plasma generation in a noble gas ionized by femtosecond two-color pulses. Here we demonstrate that, by using multiple-frequency laser pulses, one can obtain a waveform which optimizes the free electron trajectories in such a way that they reach the highest velocity at the electric field extrema. This allows to increase the THz conversion efficiency to the percent level, an unprecedented performance for THz generation in gases. Besides the analytical study of THz generation using a local current model, we perform comprehensive 3D simulations accounting for propagation effects which confirm this prediction. Our results show that THz conversion via tunnel ionization can be greatly improved with well-designed multicolor pulses.
• ### Destruction of superconductivity in disordered materials : a dimensional crossover(1408.3673)

Aug. 15, 2014 cond-mat.supr-con
The disorder-induced Superconductor-to-Insulator Transition in amorphous Nb$_{x}$Si$_{1-x}$ two-dimensional thin films is studied for different niobium compositions $x$ through a variation of the sample thickness $d$. We show that the critical thickness $d_c$, separating a superconducting regime from an insulating one, increases strongly with diminishing $x$, thus attaining values of over 100 {\AA}. The corresponding phase diagram in the $(d, x)$ plane is inferred and related to the three-dimensional situation. The two-dimensional Superconductor-to-Insulator Transition well connects with the three-dimensional Superconductor-to-Metal Transition.
• ### The fundamental solution of the unidirectional pulse propagation equation(1403.4444)

The fundamental solution of a variant of the three-dimensional wave equation known as "unidirectional pulse propagation equation" (UPPE) and its paraxial approximation is obtained. It is shown that the fundamental solution can be presented as a projection of a fundamental solution of the wave equation to some functional subspace. We discuss the degree of equivalence of the UPPE and the wave equation in this respect. In particular, we show that the UPPE, in contrast to the common belief, describes wave propagation in both longitudinal and temporal directions, and, thereby, its fundamental solution possesses a non-causal character.
• ### Purification of molybdenum oxide, growth and characterization of medium size zinc molybdate crystals for the LUMINEU program(1312.3515)

Dec. 12, 2013 nucl-ex, physics.ins-det
The LUMINEU program aims at performing a pilot experiment on neutrinoless double beta decay of 100Mo using radiopure ZnMoO4 crystals operated as scintillating bolometers. Growth of high quality radiopure crystals is a complex task, since there are no commercially available molybdenum compounds with the required levels of purity and radioactive contamination. This paper discusses approaches to purify molybdenum and synthesize compound for high quality radiopure ZnMoO4 crystal growth. A combination of a double sublimation (with addition of zinc molybdate) with subsequent recrystallization in aqueous solutions (using zinc molybdate as a collector) was used. Zinc molybdate crystals up to 1.5 kg were grown by the low-thermal-gradient Czochralski technique, their optical, luminescent, diamagnetic, thermal and bolometric properties were tested.
• ### Latest Progress on the QUBIC Instrument(1307.5701)

Nov. 12, 2013 astro-ph.CO, astro-ph.IM
QUBIC is a unique instrument that crosses the barriers between classical imaging architectures and interferometry taking advantage from both for high sensitivity and systematics mitigation. The scientific target is the detection of the primordial gravitational waves imprint on the Cosmic Microwave Background which are the proof of inflation, holy grail of modern cosmology. In this paper, we show the latest advances in the development of the architecture and the sub-systems of the first module of this instrument to be deployed in Dome Charlie Concordia base - Antarctica in 2015.
• ### Axion searches with the EDELWEISS-II experiment(1307.1488)

July 4, 2013 hep-ph, astro-ph.CO
We present new constraints on the couplings of axions and more generic axion-like particles using data from the EDELWEISS-II experiment. The EDELWEISS experiment, located at the Underground Laboratory of Modane, primarily aims at the direct detection of WIMPs using germanium bolometers. It is also sensitive to the low-energy electron recoils that would be induced by solar or dark matter axions. Using a total exposure of up to 448 kg.d, we searched for axion-induced electron recoils down to 2.5 keV within four scenarios involving different hypotheses on the origin and couplings of axions. We set a 95% CL limit on the coupling to photons $g_{A\gamma}<2.13\times 10^{-9}$ GeV$^{-1}$ in a mass range not fully covered by axion helioscopes. We also constrain the coupling to electrons, $g_{Ae} < 2.56\times 10^{-11}$, similar to the more indirect solar neutrino bound. Finally we place a limit on $g_{Ae}\times g_{AN}^{\rm eff}<4.70 \times 10^{-17}$, where $g_{AN}^{\rm eff}$ is the effective axion-nucleon coupling for $^{57}$Fe. Combining these results we fully exclude the mass range $0.91\,{\rm eV}<m_A<80$ keV for DFSZ axions and $5.73\,{\rm eV}<m_A<40$ keV for KSVZ axions.
• ### Background studies for the EDELWEISS dark matter experiment(1305.3628)

May 15, 2013 hep-ex, physics.ins-det
The EDELWEISS-II collaboration has completed a direct search for WIMP dark matter using cryogenic Ge detectors (400 g each) and 384 kg$\times$days of effective exposure. A cross-section of $4.4 \times 10^{-8}$ pb is excluded at 90% C.L. for a WIMP mass of 85 GeV. The next phase, EDELWEISS-III, aims to probe spin-independent WIMP-nucleon cross-sections down to a few $\times10^{-9}$ pb. We present here the study of gamma and neutron background coming from radioactive decays in the set-up and shielding materials. We have carried out Monte Carlo simulations for the completed EDELWEISS-II setup with GEANT4 and normalised the expected background rates to the measured radioactivity levels (or their upper limits) of all materials and components. The expected gamma-ray event rate in EDELWEISS-II at 20-200 keV agrees with the observed rate of 82 events/kg/day within the uncertainties in the measured concentrations. The calculated neutron rate from radioactivity of 1.0-3.1 events (90% C.L.) at 20-200 keV in the EDELWEISS-II data together with the expected upper limit on the misidentified gamma-ray events ($\le0.9$), surface betas ($\le0.3$), and muon-induced neutrons ($\le0.7$), do not contradict 5 observed events in nuclear recoil band. We have then extended the simulation framework to the EDELWEISS-III configuration with 800 g crystals, better material purity and additional neutron shielding inside the cryostat. The gamma-ray and neutron backgrounds in 24 kg fiducial mass of EDELWEISS-III have been calculated as 14-44 events/kg/day and 0.7-1.4 events per year, respectively. The results of the background studies performed in the present work have helped to select better purity components and improve shielding in EDELWEISS-III to further reduce the expected rate of background events in the next phase of the experiment.

format: plain


text:
# 1680

Previous  ... Next

## Number

$1680$ (one thousand, six hundred and eighty) is:

$2^4 \times 3 \times 5 \times 7$

The $17$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$, $840$, $1260$:
$\map {\sigma_0} {1680} = 40$

The $17$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$, $840$, $1260$:
$\dfrac {\map {\sigma_1} {1680} } {1680} = \dfrac {5952} {1680} \approx 3 \cdotp 543$

The $24$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $645$, $736$, $833$, $936$, $1045$, $1160$, $1281$, $1408$, $1541$:
$1680 = \ds \sum_{k \mathop = 1}^{24} \paren {6 k - 5} = 24 \paren {3 \times 24 - 2}$

### Arithmetic Functions on $1680$

 $\ds \map {\sigma_0} { 1680 }$ $=$ $\ds 40$ $\sigma_0$ of $1680$ $\ds \map {\sigma_1} { 1680 }$ $=$ $\ds 5952$ $\sigma_1$ of $1680$


text:
# 4.03 Units of area

Worksheet
Units of area
1

Consider the following units of measure for area:

• Hectares

• Square centimetres

• Square metres

• Square kilometres

Choose the most appropriate unit of measure for the following areas:

a

Postage stamp

b

Bathroom wall

c

Australia

d

Matchbox

e

Exercise book

f

Street

g

School playground

h

Pencil case

i

Japan

j

Cabinet

k

City

l

Television

m

Kitchen

n

Farm

o

Classroom

p

Netball court

q

Table

r

National park

s

Door

2

Convert the following to the unit indicated:

a

3\text{ m}^{2} to square centimetres

b

7\text{ m}^{2} to square centimetres

c

0.56 \text{ km}^{2} to square metres

d

810 \text{ mm}^{2} to square centimetres

e

34\,000 \text{ mm}^{2} to square metres

f

790\,000\,000\text{ cm}^{2} to square kilometres

g

6 \text{ ha} to square metres

h

10.4 \text{ ha} to square metres

i

20\,000 \text{ m}^{2} to hectares

j

54\,800 \text{ m}^{2} to hectares

3

Find the area of a square of side 290 \text{ cm}, in square metres.

Estimating area
4

Estimate the total area of the artwork above the sofa, in square centimetres, if the sofa is 1.9 \text{ m} in length.

5

The following garden bed has three rows of four plants. Each plant is spaced about 24 \text{ cm} apart, and sit 12 \text{ cm} away from the edge of the garden bed:

Estimate the area of the garden bed in square centimetres.

6

Each square on the grid has an area of 9 \text{ mm}^{2} . Estimate for the area of the curved shape shown:

7

Each square on the grid has an area of 4 \text{ mm}^{2}. Estimate the area of the curved shape shown:

8

Estimate the area of the shape shown if each axis is measured in metres:

9

Estimate the area of the butterfly if each axis is measured in millimetres:

10

These road signs are pictured with relative sizings. If the GIVE WAY sign is an equilateral triangle with a side length of 75 \text{ cm}, estimate the area of the BICYCLES EXCEPTED sign.

11

Estimate the areas of the following shapes:

a
b
c
d
12

Name an object that has an area of approximately 1 \text{ m}^{2}.

13

Name an object that has an area of approximately 500\text{ cm}^{2}.

Applications
14

A park has an area measuring 19 acres. Given that 1 acre is approximately 4000 \text{ m}^2, find the area of the park in square metres.

15

Rectangular farms around Australia were measured and their dimensions, in metres, are recorded in the following table:

a

Find the area of each farm in square metres.

b

Which farms have an area of more than 1 \text{ ha}?

c

Which farms have an area of less than 1 \text{ ha}?

d

Which farm has an area of exactly 1 \text{ ha}?

16

A property covers an area of 4.5 \text{ ha}. Given that 1 \text{ ha} is 10\,000 \text{ m}^2, calculate the following:

a

The area of the property in square metres.

b

The area of the property in acres, given that 1 acre is approximately 4000 \text{ m}^2.

17

A large cattle station covers an area of 320 acres. Given that 1 acre is approximately 0.4 \text{ ha}, and 1 \text{ ha} is 10\,000 \text{ m}^2, calculate the following:

a

The area of the cattle station in hectares.

b

The area of the cattle station in square kilometres.

18

A farm covers an area of 51\,000 \text{ m}^2. Given that 1 \text{ ha} is 10\,000 \text{ m}^2, and 1 acre is approximately 4000 \text{ m}^2, calculate the following:

a

The area of the farm in hectares.

b

The area of the farm in acres. Round your answer to two decimal places.

19

Bob owns a paddock with an area of 290 acres. He wants to advertise his eggs as free range, with no more than 400 chickens per hectare. Given that 1 acre is approximately 0.4 \text{ ha}, find the largest amount of chickens that Bob can keep in his paddock.

20

A farm covers an area measuring 340 acres. Given that 1 acre is approximately 0.4 \text{ ha}, and 1 \text{ ha} is 10\,000 \text{ m}^2, calculate the following:

a

The area of the farm in hectares.

b

The area of the farm in square metres.

21

American football is played on a field with an area of 12\,000 \text{ yd}^2.

a

Given that 1 \text{ yd}^2 is approximately 0.836\text{ m}^2, find the area of an American football field in square metres.

b

The MCG oval has an area of 17\,700\text{ m}^2. How much smaller than the MCG oval is an American football field? Give your answer in square metres.

22

Manhattan, the most densely populated borough of New York City, has an area of 22.8\text{ mi}^2. If 1\text{ mi}^2 is approximately 2.6 \text{ km}^2, how large is Manhattan in square kilometres?

23

The state of California is roughly 164\,000 \text { mi}^2 in area. Given that 1 \text { mi}^2 is approximately 2.6 \text{ km}^2:

a

Find the area of California in square kilometres.

b

The state of New South Wales has an area of approximately 809\,000 \text{ km}^2. How much larger is New South Wales than California?

24

An official NBA court covers 4700 \text{ ft}^2 in area. Given that 1 \text{ m}^2 is approximately 10.8 \text{ ft}^2, find the area of an official basketball court in square metres. Round your answer to three decimal places.

### Outcomes

#### 1.3.5

choose and use appropriate metric units of area, their abbreviations and conversions between them

#### 1.3.6

estimate the areas of different shapes

#### 1.3.7

convert between metric units of area and other area units


text:
# Axiom of Union?

I'm reading Comprehensive Mathematics for Computer Scientists 1. On the second chapter: Axiomatic Set Theory.

He first states the axiom of the empty set, the axiom of equality and then he proceeds to the axiom of union:

Axiom 3 (Axiom of Union) If $a$ is a set, then there is a set:

$\{$$x | there exists an element b\in a such that x\in b$$\}$.

This set is denoted by $\bigcup a$ and is called the union of $a$.

Notation 2 If a = {b,c}. or a = {b,c,d}, respectively, one also writes b $\cup$ c, or b $\cup$ c $\cup$ d, respectively, instead of $\cup$a

I've learned the definition of Union while I was in school, but it wasn't with axioms, they just gave an intuitive example:

$a=\{1,2,3\}$

$b=\{4,5\}$

$a\bigcup b=\{1,2,3,4,5\}$

I can't see how the notion of this intuitive example happens on the axiom of union. In my example, it's easy to understand because there's a mention to another set, where's the mention in this axiom?

-
In ZFC, any element of a set is itself a set, so the interpretation of $\cup a$ is the union of all the sets in $a$. – Kris Aug 24 '12 at 20:54
The exposition in that book is rather terse, especially for a book addressed to non-mathematicians (something noted in its Amazon reviews.) There are better examples of the axiom of union at work in Hrbacek and Jech, p. 10, although unlike everything in the answers shown below, the examples of Hrbacek and Jech use "pure" set theory i.e. without urelements like $a$ or $b$. (In layman's terms, this means only the empty set and braces are used to build sets.) It's somewhat insightful to see how the axiom works out in that context as well. – Fizz Apr 12 '15 at 3:16
Also if you do have urelements (aka atoms) other than the empty set, then the axiom of union loses the "top level" ones. E.g. $\bigcup \{a,\{b\}\}$ is just $\{b\}$. This is one of the troubles with urelements and why the axiom looks strange with urelements. Hat tip to Tourlakis' book for mentioning this. – Fizz Apr 12 '15 at 4:06

The connection between your example and the more general definition is that $\bigcup\{a,b\}=a\cup b$. Written out in all its gory details, this is

$$\bigcup\Big\{\{1,2,3\},\{4,5\}\Big\}=\{1,2,3\}\cup\{4,5\}=\{1,2,3,4,5\}\;.$$

Let’s check that against the definition:

\begin{align*} &\bigcup\Big\{\{1,2,3\},\{4,5\}\Big\}\\ &\qquad=\left\{x:\text{there exists an element }y\in\Big\{\{1,2,3\},\{4,5\}\Big\}\text{ such that }x\in y\right\}\\ &\qquad=\Big\{x:x\in\{1,2,3\}\text{ or }x\in\{4,5\}\Big\}\\ &\qquad=\{1,2,3\}\cup\{4,5\}\\ &\qquad=\{1,2,3,4,5\}\;. \end{align*}

Take a slightly bigger example. Let $a,b$, and $c$ be any sets; then

\begin{align*} \bigcup\{a,b,c\}&=\Big\{x:\text{there exists an element }y\in\{a,b,c\}\text{ such that }x\in y\Big\}\\ &=\{x:x\in a\text{ or }x\in b\text{ or }x\in c\}\\ &=a\cup b\cup c\;. \end{align*}

One more, even bigger: for $n\in\Bbb N$ let $A_n$ be a set, and let $\mathscr{A}=\{A_n:n\in\Bbb N\}$. Then

\begin{align*} \bigcup\mathscr{A}&=\Big\{x:\text{there exists an }n\in\Bbb N\text{ such that }x\in A_n\Big\}\\ &=\{x:x\in A_0\text{ or }x\in A_1\text{ or }x\in A_2\text{ or }\dots\}\\ &=A_0\cup A_1\cup A_2\cup\dots\\ &=\bigcup_{n\in\Bbb N}A_n\;. \end{align*}

-
If there exists an element $y\in\Big\{\{1,2,3\},\{4,5\}\Big\}$, then shouldn't we have a $\Big\{\{1,2,3\},\{4,5\},y\Big\}$? – Voyska Aug 24 '12 at 18:15
@Gustavo: No: $y$ is a dummy name used here to stand for any member of the set $\Big\{\{1,2,3\},\{4,5\}\Big\}$. Here the possible values of $y$ are $\{1,2,3\}$ and $\{4,5\}$. – Brian M. Scott Aug 24 '12 at 18:17
@Gustavo: Yes, $x$ in expressions like $\{x:\text{something}\}$ is also a dummy variable; it can stand for anything that satisfies the condition $\text{something}$. – Brian M. Scott Aug 24 '12 at 18:34
Then I have to choose one of the sets on: $\{\{1,2,3 \},\{4,5 \}\}$ and then choose an element inside the chosen one? – Voyska Aug 24 '12 at 18:37
Yep. Your last comment seems to tell me that I should consider all possible options, isn't it? – Voyska Aug 24 '12 at 18:38

Let $A=\{a,b\}$ (the set whose only elements are $a$ and $b$). Then the union of $a$ and $b$ that you described is what the Axiom of Union produces from $A$.

Remark: Informally, let $A$ be a set whose elements are a bunch of plastic bags with stuff in them (so $A$ is a set of sets). Then the set produced by the Axiom of Union from $A$ dumps the stuff contained in the bags into a single bag. (Duplicates are thrown away.)

-
Oh, then $\bigcup a$ acts like a variable. When he says: "There is a set..." he refers to a set that have no name, it's only $\{$x | such that... $\}$ and not $z=\{$x | such that... $\}$, Then I thought that this nameless set were implicit here $\bigcup a$, if it had a name $z$, it would be written like: $z\bigcup a$. Is this right? – Voyska Aug 24 '12 at 18:02
@GustavoBandeira: I am having trouble understanding what you mean. Here $\cup$ acts as a unary operator (function). If you apply it to the set $\{a,b,c,d\}$ of sets it produces $a\cup b\cup c\cup d$. It operates similarly on an infinite collection $\{a_1,a_2,a_3,\dots\}$ of sets. – André Nicolas Aug 24 '12 at 18:11
I wasn't aware that I could write it like LISP. – Voyska Aug 24 '12 at 18:59
@GustavoBandeira: It is a common notation. In principle, all we use is $\in$ and logical symbols, but in doing doing set theory it is then useful (indeed almost necessary!) to introduce abbreviations for important constructions. – André Nicolas Aug 24 '12 at 19:56

When we write $a\cup b$ we actually mean $\bigcup\{a,b\}$. This is a shorthand instead of writing long formulas every time we want to talk about the union of two sets.

-
Yep. Same as I commented here. – Voyska Aug 24 '12 at 18:04
I've read your comment with attention now. This reminds me of LISP where you can write (+ 2 3). – Voyska Aug 24 '12 at 19:41
@Gustavo: Think of $\bigcup$ as a LISP function "union": $$(\textrm{union }a\ b\ \ldots)$$ It takes a list of sets and returns their union. The $a\cup b$ notation is a bit like C syntax. – Asaf Karagila Aug 24 '12 at 20:35

Think of $a$ as a set (or collection, if you like) of other sets. Then $\bigcup a$ is the union of all these sets. So, for instance, in your example:

$$\bigcup \lbrace\lbrace 1,2,3\rbrace,\lbrace 4,5\rbrace\rbrace = \lbrace 1,2,3,4,5\rbrace$$

You may think of $A\cup B$ as shorthand for $\bigcup \lbrace A,B\rbrace$.

-
Yep, it's the same as I pointed here. – Voyska Aug 24 '12 at 18:03

This axiom talks about a set of sets.

This is because the axiom states $b\in a$ and $x\in b$: $x$ in $b$ tells you that $b$ is a set (and is an element of $a$).

For example: $a=\{\{1\},\{2,3\}\}$ then the axiom states that $\{1\}\cup\{2,3\}=\{1,2,3\}$ exists.

-


text:
Volume 2, Issue 0
Numerical Simulation and Analysis of Migration-Accumulation of Oil Resources

Int. J. Numer. Anal. Mod., 2 (2005), pp. 68-82.

Published online: 2005-11

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• Abstract

Numerical simulation of migration-accumulation of oil resources in porous media is to describe the history of oil migration and accumulation in basin evolution. It is of great value to the evaluation of oil resources and to the determination of the location and amount of oil deposits. This thesis puts forward a mathematical model, a careful parallel operator splitting-up implicit iterative scheme, parallel arithmetic program, parallel arithmetic information and alternating-direction mesh subdivision. For the actual situation of Tanhai region of Shengli Petroleum Field, our numerical simulation test results and the actual conditions are coincident. For the model problem (nonlinear coupled system) optimal order estimates in $l^2$ norm are derived to determine the errors. We have successfully solved the difficult problem in the fields of permeation fluid mechanics and petroleum geology.

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• TXT
@Article{IJNAM-2-68, author = {}, title = {Numerical Simulation and Analysis of Migration-Accumulation of Oil Resources}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2005}, volume = {2}, number = {0}, pages = {68--82}, abstract = {

Numerical simulation of migration-accumulation of oil resources in porous media is to describe the history of oil migration and accumulation in basin evolution. It is of great value to the evaluation of oil resources and to the determination of the location and amount of oil deposits. This thesis puts forward a mathematical model, a careful parallel operator splitting-up implicit iterative scheme, parallel arithmetic program, parallel arithmetic information and alternating-direction mesh subdivision. For the actual situation of Tanhai region of Shengli Petroleum Field, our numerical simulation test results and the actual conditions are coincident. For the model problem (nonlinear coupled system) optimal order estimates in $l^2$ norm are derived to determine the errors. We have successfully solved the difficult problem in the fields of permeation fluid mechanics and petroleum geology.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/948.html} }
TY - JOUR T1 - Numerical Simulation and Analysis of Migration-Accumulation of Oil Resources JO - International Journal of Numerical Analysis and Modeling VL - 0 SP - 68 EP - 82 PY - 2005 DA - 2005/11 SN - 2 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/948.html KW - AB -

Numerical simulation of migration-accumulation of oil resources in porous media is to describe the history of oil migration and accumulation in basin evolution. It is of great value to the evaluation of oil resources and to the determination of the location and amount of oil deposits. This thesis puts forward a mathematical model, a careful parallel operator splitting-up implicit iterative scheme, parallel arithmetic program, parallel arithmetic information and alternating-direction mesh subdivision. For the actual situation of Tanhai region of Shengli Petroleum Field, our numerical simulation test results and the actual conditions are coincident. For the model problem (nonlinear coupled system) optimal order estimates in $l^2$ norm are derived to determine the errors. We have successfully solved the difficult problem in the fields of permeation fluid mechanics and petroleum geology.

Y. Yuan. (1970). Numerical Simulation and Analysis of Migration-Accumulation of Oil Resources. International Journal of Numerical Analysis and Modeling. 2 (0). 68-82. doi:
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# How do you write 90% as a fraction?

Nov 17, 2016

So we know that percentage, is a fraction itself.

90%=90/100

Simplify this and it becomes $\frac{9}{10}$.

Hope this helps :)

Jun 24, 2017

So, just to be clear,
The percent symbol(%), means upon 100, so here it is 90%, which will mean $\frac{90}{100}$, which, lowered to its lowest terms, will be,
$\frac{9}{10}$


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# On Definition of Random Variables

Given a probability space $$(\Omega, \mathcal{F}, P)$$, a random variable X is defined to be a function from $$\Omega$$ to $$\mathbb{R}$$ such that for every $$a \in \mathbb{R}$$, the set $$\{X \leq a\} := \{ω \in \Omega : X(\omega) ≤ a\}$$ belongs to $$\mathcal{F}$$.

From the axiomatic definition of a probability space, for every $$a \in \mathbb{R}$$, if the set $$\{X \leq a\}$$ belongs to $$\mathcal{F}$$, the set $$\{X > a\}$$ must also belong to $$\mathcal{F}$$. Consequently, the set $$\{a < X \leq b\}$$ must also belong to $$\mathcal{F}$$.

I want to show that the set $$\{a \leq X < b\}$$ also belongs to $$\mathcal{F}$$ as well, so I can generalize it to any other set, but I have not managed to prove it.

Anyone has an idea on how to do it?

• This might be a way to think about it, although not necessarily the simplest way: we can find an infinite sequence $\{a_i\}$ that converges to $a$ from below. Therefore, $\bigcup_{i=1}^\infty \{X\le a_i\}=\{X<a\}$. This set is also in $\mathcal{F}$, because $\mathcal{F}$ is $\sigma$-algebra. Jun 15 at 1:08

## 3 Answers

As @whoknowsnot has shown, for every $$a \in \mathbb R$$, the event $$\{X< a\}\in \mathcal F$$, and since $$\{X\leq a\}\in \mathcal F$$ by definition, the difference of these two events, namely $$\{X = a\}$$, is also in $$\mathcal F$$. Hence, from the result that you have already proven, namely that $$\{a < X \leq b\} \in \mathcal F$$, it is easy to deduce that \begin{align} \{a < X < b\} &\in \mathcal F,\\ \{a \leq X < b\} &\in \mathcal F,\\ \{a \leq X \leq b\} &\in \mathcal F, \end{align} just by adding in or taking away the singleton events $$\{X = a\}$$ and $$\{X = b\}$$.

To quote from Wikipedia, a $$\sigma$$-algebra is defined as

1. $$\Omega$$ is in $$\mathcal F$$, and $$\Omega$$ is considered to be the universal set in the following context.
2. $$\mathcal F$$ is closed under complementation: If $$A$$ is in $$\mathcal F$$, then so is its complement, $$A^c$$.
3. $$\mathcal F$$ is ''closed under countable unions'': If $$A_1$$, $$A_2,\ldots$$ are in $$\mathcal F$$, then so is $$A=A_1 ∪ A_2 ∪ \cdots$$

From these properties, it follows that the σ-algebra is also closed under countable intersection (by applying De Morgan's laws).

You can write the desired set using only negation and countable unions of the foundational sets as:

\begin{align} \{ a \leq X < b \} &= \{ a \leq X \leq b \} - \overline{ \{ X \leq b \}} \\[6pt] &= \{ X \leq b \} - \bigcup_{n=1}^\infty \bigg\{ X \leq a - \frac{1}{n} \bigg\} - \overline{ \{ X \leq b \}}. \\[6pt] \end{align}

Since the sigma-field $$\mathcal{F}$$ is closed under negation, countable unions and set difference (based on the foundation sets) this is sufficient to prove that $$\{ a \leq X < b \} \in \mathcal{F}$$.


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# Exploring the relationship between alexithymia and empathy : the role of emotion recognition and metacognitive ability

Spicer, LC 2018 , 'Exploring the relationship between alexithymia and empathy : the role of emotion recognition and metacognitive ability', Honours thesis, University of Tasmania.

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Alexithymic traits have been consistently found to be associated with deficits in social functioning. These social deficits in individuals with alexithymic traits have been suggested to be related to the reduced levels of empathy in those with alexithymic traits. Factors underlying this relationship, however, are presently not understood. Such an understanding may provide insight into addressing the empathy deficits seen in alexithymic individuals. This study aimed to support prior studies that indicate a relationship between cognitive and affective alexithymic traits and cognitive and affective domains of empathy, respectively. A preliminary investigation of the impact of emotion recognition ability (anger, fear, sadness) and metacognitive ability on the relationship between alexithymic traits and empathy was also of interest. One hundred and twenty participants aged between 18 and 55 years (68 females; M = 24.95, SD = 7.19) completed the Toronto Alexithymia Scale – 20, the Interpersonal Reactivity Index (IRI), the Emotion Recognition Task (ERT) and provided confidence ratings for each presented emotion of the ERT to assess metacognitive ability. The study identified negative relationships between cognitive alexithymic traits, and cognitive empathy across the emotions of anger, fear, and sadness. A moderated mediation effect for this relationship was also found for the emotion sadness. The findings suggest poor emotion recognition of sadness negatively impacts the relationship between cognitive alexithymic traits and cognitive empathy, but only for those with low levels of metacognitive ability. No significant relationships were found between affective alexithymic traits and affective empathy. These findings upon further testing in a clinical population may contribute to addressing empathy deficits seen in alexithymic individuals, in order to improve their social functioning.


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# Find the common ratio of the geometric series with the sum and the first term

Given:
Geometric Series Sum = 39
First Term = 3
number of terms = 3

*I do not know how to use special characters. I hope you can understand this. I need a formula for looking the common ratio of a geometric series. To make it more clear, the common ratio is 3.

• The solutions for this restricted problem are $r=3$ and $r=-4$. Just need to solve $1+r+r^2=13$. Sum of geometric formula is not useful here. – André Nicolas Feb 26 '13 at 5:17
• @AndréNicolas, I am very sure that the sum is correct and it should be included. (Sorry for unclear explanations) – clixbrigidxterx Feb 26 '13 at 5:25
• I meant that to solve this problem, you do not need the formula $a\frac{r^n-1}{r-1}$ for the sum of the GP $a+ar+\cdots +ar^{n-1}$. Since there are only $3$ terms, want $3+3r+3r^2=39$, which simplifies to $r^2+r-12=0$. A nice quadratic, it even factors. – André Nicolas Feb 26 '13 at 5:35
• 3,-12,48 works as does 3,9,27 so there are a pair of answers here. – JB King Feb 26 '13 at 5:38

If you have an infinite geometric series with first term $a$ and common ratio $r$ (with $|r|\lt1$) then the sum $s$ is $$s={a\over1-r}$$

In your problem, you know $s$ and $a$ --- can you solve for $r$?

EDIT: OP has clarified that the number of terms is $3$.

That means the terms are $3$, $3r$, and $3r^2$. And they add up to $39$. So --- can you work out $r$ from this information?

• Yes, I need the formula and how did you create that formula from the original geometric series formula. – clixbrigidxterx Feb 26 '13 at 5:05
• @clixbrigidxterx: If $a$ is the first term and $r$ is the ratio, you have $a(1+r+r^2)=39$. It seems likely that $a,r$ are integers, so factor $39$ and try them all. There aren't many. – Ross Millikan Feb 26 '13 at 5:08
• @clixbrigidxterx: if you know $r=3$ what is $1+r+r^2$? What do you know, it divides into $39$ – Ross Millikan Feb 26 '13 at 5:09
• @clixbrigidxterx Or you can compute the roots of the quadratic. – Julien Feb 26 '13 at 5:12
• @clixbrigidxterx did you need to construct the formula from scratch or do you just need to solve for r? I think Gerry explained it pretty well. Your first term is a, second term is the first term multiplied by r (ar), third term is the second multiplied by r ($ar^2$) etc. It's a good exercise to check that the formula from my answer holds for finite geometric series, and the formula Gerry wrote for his holds for infinite geometric series – user62228 Feb 26 '13 at 5:12

You have $a+ar+ar^2=39$. Do you understand how this comes from what you asked? Then $a\frac {r^3-1}{r-1}=39$ as shown in Wikipedia on geometric progression. Given that $r=3$, we get $a\frac {26}2=39$ so $a=3$ and we have $3+9+27=39$.

• I want to see the process it gone through. Please, do not use the answer (common ratio) to substitute $r$. I only include this answer so you can prove that the answer is 3. – clixbrigidxterx Feb 26 '13 at 5:30
• @clixbrigidxterx: See André Nicolas' comment. He has it very simply for a 3 term progression. Then JB King solves the quadratic. For more general problems, you need to read on geometric progressions, which derive the sum of the series formula that we have all quoted. – Ross Millikan Feb 26 '13 at 14:04

Finite progression with $\,n\,$ elements:

$$39=S_n=3\frac{q^n-1}{q-1}\Longrightarrow \frac{q^n-1}{q-1}=13\ldots\text{we need to know}\,\,\,n$$

Infinite descending geometric series:

$$39=S=\frac{3}{1-q}\Longrightarrow 1-q=\frac{1}{13}\Longrightarrow q=\frac{12}{13}$$

You have the formula $$S_n=\frac{a*(1-r^n)}{1-r}$$ In your case, n=3, a=3 and $S_n$=39. Now solve for r.

• @user6228, can you make show me how can I do it? I am stuck of finding the ratio in the formula because of exponents and some factors. – clixbrigidxterx Feb 26 '13 at 5:14


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# Bold, Italic and underlined.

• This bold.
• This is italic..
• This is underlined..

# Numbered list

1. Mathematics
1. Algebra
2. Matrix
2. English
3. Science

## Bullet

• Mathematics
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• ==A good search engine==

### Koforidua Polytechnic

Learning4Content[[|]]

## My Content Development Projects

Assignment Type Due Date

Objectives
 Use SQC techniques to monitor production process

Objectives
 At the end of the module, students would be able to use Operations Research Techniques to analyse business data

# Assignment

 See guidelines for submitting assignment

# Reflection

 Go over the previous work and plan for the next activity |

| Historical Background |Sources of Variation | Statistical Process Control | Statistical Process Control Methods | Control Charts for Variables | Control Charts for Attributes |C-Charts |Process Capability|Acceptance Sampling|

Which statement is Correct?
 A. 2 + 3 = 5 Read the question again! B. 4 / 9 = 1 WRONG! Try again. C. 1 + 1 = 2 Read the question again! D. Both A and C Correct!

## My optional community service (learning contract) project

 Agreement By signing this optional learning contract I will try to complete my training in basic wiki editing skills to achieve the status of a Wikibuddyabd proceed to WikiArtisan. In return for this free training opportunity, I will give the gift of knowledge by donating or developing at least one free content resource licensed under a CC-BY-SA or CC-BY license which can be used by myself (and others) on WikiEducator. I shall also work in collaboration with Alison Snieckus on the math community project, at WikiEducator:Mathematics Community Signature Godfred Kwame Abledu --Godfredabledu Brief description of project Target date for completion .

Today is: Monday ,3 October 2022. GMT Time is: 17:53.Template:CALENDER


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# [Tex/LaTex] Horner scheme with a tabular

tables

I know there exists a standard code for a Horner scheme. But I would like to get the following tabular.

I've got the next code but I can't find how I get that little vertical line at the right bottom.

\begin{tabular}{c|cccc}
& 3 & 2 & -5 & -10 \\
&   &   &    &     \\ \hline
&   &   &    &
\end{tabular}

I've got another question about this. The number 11 is now shifted to the left, but I want that it's under 16.

\begin{tabular}{c|cccc}
& 3 & 2 & -5 & -10 \\
2 & $\downarrow$  & 6  & 16    & 22   \\ \hline
& 3  & 8  &  11   \multicolumn{1}{c|}{}  & 12
\end{tabular}

\documentclass{article}
\begin{document}
$\begin{array}{c|cccc} & 3 & 2 & -5 & -10 \\ & & & & \\ \hline & & & \multicolumn{1}{c|}{} & \end{array}$
\end{document}

If you want it to be thick,

\documentclass{article}
\usepackage{array}
\begin{document}
$\begin{array}{c|cccc} & 3 & 2 & -5 & -10 \\ & & & & \\ \hline & & & & \multicolumn{1}{!{\vrule width 1.5pt}c}{} \end{array}$
\end{document}


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# 9.5: Coordinate Covalent Bond

Remember when you were younger, and were told to share your favorite toy with your brother, sister, or friend? You probably didn't want to share, but did anyway. It likely turned out that you had more fun playing with the toy together than if you had kept it to yourself. Atoms often share electrons with other atoms that have nothing to contribute to the situation; the end result is a new structure.

## Coordinate Covalent Bonds

Each of the covalent bonds that we have looked at so far has involved each of the atoms that are bonding contributing one of the electrons to the shared pair. There is an alternate type of covalent bond in which one of the atoms provides both of the electrons in a shared pair. Carbon monoxide, $$\ce{CO}$$, is a toxic gas that is released as a byproduct during the burning of fossil fuels. The bonding between the $$\ce{C}$$ atom and the $$\ce{O}$$ atom can be thought of in the following procession:

At this point, a double bond has formed between the two atoms, with each atom providing one of the electrons to each bond. The oxygen atom now has a stable octet of electrons, but the carbon atom only has six electrons and is unstable. This situation is resolved if the oxygen atom contributes one of its lone pairs in order to make a third bond with the carbon atom.

The carbon monoxide molecule is correctly represented by a triple covalent bond between the carbon and oxygen atoms. One of the bonds is a coordinate covalent bond, a covalent bond in which one of the atoms contributes both of the electrons in the shared pair.

Once formed, a coordinate covalent bond is the same as any other covalent bond. It is not as if the two conventional bonds in the $$\ce{CO}$$ molecule are stronger or different in any other way than the coordinate covalent bond.

## Summary

• Coordinate covalent bonds can form when one atom provides a lone pair of electrons to the bond.
• Coordinate covalent bonds are as strong as other covalent bonds.


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# 2.4.6: Exercises

## Analysis

1. In the circuit of Figure 5.6.1 , which individual resistors are strictly in series and which are in parallel?

Figure 5.6.1

2. In the circuit of Figure 5.6.2 , which individual resistors are strictly in series and which are in parallel?

Figure 5.6.2

3. In the circuit of Figure 5.6.3 , which individual resistors are strictly in series and which are in parallel?

Figure 5.6.3

4. In the circuit of Figure 5.6.4 , which individual resistors are strictly in series and which are in parallel?

Figure 5.6.4

5. In the circuit of Figure 5.6.5 , which individual resistors are strictly in series and which are in parallel?

Figure 5.6.5

6. In the circuit of Figure 5.6.6 , which individual resistors are strictly in series and which are in parallel?

Figure 5.6.6

7. In the circuit of Figure 5.6.7 , which individual resistors are strictly in series and which are in parallel?

Figure 5.6.7

8. In the circuit of Figure 5.6.8 , which individual resistors are strictly in series and which are in parallel?

Figure 5.6.8

9. Determine the equivalent resistance of the network shown in Figure 5.6.1 (i.e., as if an ohmmeter is connected to the two terminals).

10. Determine the equivalent resistance of the network shown in Figure 5.6.2 .

11. Determine the equivalent resistance of the network shown in Figure 5.6.3 .

12. Determine the equivalent resistance of the network shown in Figure 5.6.4 .

13. Determine the equivalent resistance of the network shown in Figure 5.6.5 .

14. Determine the equivalent resistance of the network shown in Figure 5.6.6 .

15. Determine the equivalent resistance of the network shown in Figure 5.6.7 .

16. Determine the equivalent resistance of the network shown in Figure 5.6.8 .

17. For the circuit of Figure 5.6.9 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

Figure 5.6.9

18. For the circuit of Figure 5.6.9 , find the current through each resistor.

19. For the circuit of Figure 5.6.10 , find the current through each resistor.

Figure 5.6.10

20. For the circuit of Figure 5.6.10 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

21. For the circuit of Figure 5.6.11 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

Figure 5.6.11

22. For the circuit of Figure 5.6.11 , find the current through each resistor.

23. For the circuit of Figure 5.6.12 , find the current through each resistor.

Figure 5.6.12

24. For the circuit of Figure 5.6.12 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

25. In the circuit of Figure 5.6.13 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

Figure 5.6.13

26. In the circuit of Figure 5.6.13 , find the current through each resistor.

27. In the circuit of Figure 5.6.14 , find the current through each resistor.

28. In the circuit of Figure 5.6.14 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

Figure 5.6.14

29. For the circuit of Figure 5.6.15 , find voltages $$V_b$$, $$V_c$$ and $$V_{cb}$$.

Figure 5.6.15

30. For the circuit of Figure 5.6.15 , find the current through the 470 $$\Omega$$ and 3.6 k$$\Omega$$ resistors along with the source current.

31. For the circuit of Figure 5.6.16 , find the current through the 2 k$$\Omega$$ and 5.1 k$$\Omega$$ resistors along with the source current.

Figure 5.6.16

32. For the circuit of Figure 5.6.16 , find voltages $$V_c$$, $$V_b$$, $$V_{bc}$$ and $$V_{ab}$$.

33. For the circuit of Figure 5.6.17 , find voltages $$V_a$$ and $$V_b$$.

Figure 5.6.17

34. For the circuit of Figure 5.6.17 , find the current through the 10 k$$\Omega$$ and 4 k$$\Omega$$ resistors along with the source current.

35. For the circuit of Figure 5.6.18 , find the current through the 30 k$$\Omega$$ and the left-most 36 k$$\Omega$$ resistors along with the source current.

Figure 5.6.18

36. For the circuit of Figure 5.6.18 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

37. In the circuit of Figure 5.6.18 , must the currents through the two 36 k$$\Omega$$ resistors be the same? Why/why not?

38. In the circuit of Figure 5.6.18 , must the currents through the two 48 k$$\Omega$$ resistors be the same? Further, must these currents be the same as the currents through the two 36 k$$\Omega$$ resistors Why/why not?

39. For the circuit of Figure 5.6.19 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

Figure 5.6.19

40. For the circuit of Figure 5.6.19 , find the current through the 3 k$$\Omega$$, 11 k$$\Omega$$, and both 9 k$$\Omega$$ resistors.

41. Given the circuit of Figure 5.6.20 , find the current through the 47 k$$\Omega$$, 5.1 k$$\Omega$$, and 3.9 k$$\Omega$$ resistors.

Figure 5.6.20

42. Given the circuit of Figure 5.6.20 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

43. For the circuit of Figure 5.6.21 , find voltages $$V_b$$, $$V_c$$ and $$V_d$$.

Figure 5.6.21

44. For the circuit of Figure 5.6.21 , find the current through the 6.8 k$$\Omega$$, 8.5 k$$\Omega$$, and 20 k$$\Omega$$ resistors.

45. The circuit of Figure 5.6.22 is called an R-2R ladder. Find the current through each of the 60 k$$\Omega$$ resistors.

Figure 5.6.22

46. Given the circuit of Figure 5.6.22 , find voltages $$V_b$$, $$V_c$$, $$V_d$$ and $$V_e$$. What is unique about this configuration?

47. For the circuit of Figure 5.6.23 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

Figure 5.6.23

48. For the circuit of Figure 5.6.23 , find the current through each of the resistors.

49. For the circuit of Figure 5.6.24 , find the current through each of the resistors.

Figure 5.6.24

50. For the circuit of Figure 5.6.24 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

51. Given the circuit of Figure 5.6.25 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

Figure 5.6.25

52. Given the circuit of Figure 5.6.25 , find the current through each of the resistors.

53. For the circuit of Figure 5.6.26 , find the current through the 200 $$\Omega$$, 400 $$\Omega$$, and 100 $$\Omega$$ resistors.

Figure 5.6.26

54. For the circuit of Figure 5.6.26 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

55. Given the circuit of Figure 5.6.27 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

Figure 5.6.27

56. Given the circuit of Figure 5.6.27 , find the current through the 5 k$$\Omega$$, and 6 k$$\Omega$$ resistors.

57. For the circuit of Figure 5.6.28 , find the current through the 9 k$$\Omega$$ and rightmost 82 k$$\Omega$$ resistors.

Figure 5.6.28

58. For the circuit of Figure 5.6.28 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

59. Must the current through the 82 k$$\Omega$$ resistors be identical in the circuit of 4.8.6? Why/why not?

60. Must the current through the 25 k$$\Omega$$ resistors be identical in the circuit of 4.8.6? Must they be the same as the currents through the 82 k$$\Omega$$ resistors? Why/why not?

61. For the circuit of Figure 5.6.29 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

Figure 5.6.29

62. For the circuit of Figure 5.6.29 , find the current through the 30 k$$\Omega$$, 5 k$$\Omega$$, and 1 k$$\Omega$$ resistors.

63. For the circuit of Figure 5.6.30 , find the current through the 1 k$$\Omega$$, 2.2 k$$\Omega$$, and 18 k$$\Omega$$ resistors.

Figure 5.6.30

64. For the circuit of Figure 5.6.30 , find voltages $$V_a$$, $$V_b$$ and $$V_c$$.

65. Given the circuit of Figure 5.6.31 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$.

Figure 5.6.31

66. Given the circuit of Figure 5.6.31 , find the current through the 20 k$$\Omega$$, 15 k$$\Omega$$, and 3.3 k$$\Omega$$ resistors.

67. The circuit of Figure 5.6.32 is referred to as an R-2R ladder network. Find the current through each of the 4 k$$\Omega$$ resistors.

Figure 5.6.32

68. For the circuit of Figure 5.6.32 , find voltages $$V_a$$, $$V_b$$ and $$V_{ab}$$. What is the unique characteristic of this configuration?

## Design

69. Determine a new value for the 33 $$\Omega$$ resistor in Figure 5.6.9 such that the source current is 10 mA.

70. Determine a new value for the 20 k$$\Omega$$ resistor in Figure 5.6.12 such that $$V_b$$ is 4 volts.

71. Determine a new value for the 470 $$\Omega$$ resistor in Figure 5.6.15 such that $$V_b$$ is 6 volts.

72. Determine a new value for the 1 k$$\Omega$$ resistor in Figure 5.6.14 such that the source current is 20 mA.

73. Determine a new value for the 12 k$$\Omega$$ resistor in Figure 5.6.16 such that $$V_{bc}$$ is 0 volts.

74. Given the circuit of Figure 5.6.24 , determine a new value for the 100 k $$\Omega$$ resistor such that $$V_a$$ is 75 volts.

75. Given the circuit of Figure 5.6.23 , determine a new value for the current source such that $$V_b$$ is 8 volts.

76. Determine a new value for the 2.2 k$$\Omega$$ resistor in Figure 5.6.30 such that $$V_{bc}$$ is 0 volts.

77. Given the circuit of Figure 5.6.32 , determine a new value for the current source such that $$V_e$$ is 1 volt.

## Challenge

78. Utilizing only 1 k$$\Omega$$ resistors, create a series-parallel combination that achieves 1.25 k$$\Omega$$ of total resistance.

79. Utilizing only 12 k$$\Omega$$ resistors, create a series-parallel combination that achieves 9 k$$\Omega$$ of total resistance.

80. Consider the circuit of Figure 5.6.22 . Alter the values of the two right-most 30 k$$\Omega$$ resistors such that $$V_e$$ is 1.2 volts. The remaining node voltages should be unchanged from the original circuit.

81. Alter the circuit shown in Figure 5.6.22 such that another “wrung” is added to the ladder creating a new right-most node $$f$$. If the former terminating 30 k$$\Omega$$ resistor (i.e., the vertical one) is reset to 60 k$$\Omega$$, determine the values of the resistors on the new wrung such that $$V_f$$ is 0.5 volts.

82. Given the circuit of Figure 5.6.25 , determine a new value for the 2 k $$\Omega$$ resistor such that $$V_b$$ is 12 volts.

83. Given the circuit of Figure 5.6.32 , determine new values for the resistors such that all of the node voltages are twice the value of the original circuit's node voltages.

## Simulation

84. Perform a DC simulation on the result of problem 24 to verify the node voltages.

85. Perform a DC simulation on the result of problem 25 to verify the node voltages.

86. Perform a DC simulation on the result of problem 43 to verify the node voltages.

87. Perform a DC simulation on the result of problem 46 to verify the node voltages.

88. Perform a DC simulation on the result of problem 51 to verify the node voltages.

89. Perform a DC simulation on the result of problem 68 to verify the node voltages.

90. Perform a DC simulation on the proposed solution to problem 69 to verify the new design.

91. Perform a DC simulation of the alteration presented in Challenge problem 80. Does the new design meet all of the requirements?

92. Perform a DC simulation of the alteration requested in Challenge problem 83. Does the new design meet all of the requirements?

2.4.6: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by James M. Fiore.