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PROBLEMS AND THEOREMS IN LINEAR ALGEBRA V. Prasolov Abstract. This book contains the basics of linear algebra with an emphasis on non- standard and neat proofs of known theorems. Many of the theorems of linear algebra obtained mainly during the past 30 years are usually ignored in text-books but are quite accessible for students majoring or minoring in mathematics. These theorems are given with complete proofs. There are about 230 problems with solutions. Typeset by A M S-T E X 1 CONTENTS Preface Main notations and conventions Chapter I. Determinants Historical remarks: Leibniz and Seki Kova. Cramer, L’Hospital, Cauchy and Jacobi 1. Basic properties of determinants The Vandermonde determinant and its application. The Cauchy deter- minant. Continued fractions and the determinant of a tridiagonal matrix. Certain other determinants. Problems 2. Minors and cofactors Binet-Cauchy’s formula. Laplace’s theorem. Jacobi’s theorem on minors of the adjoint matrix. The generalized Sylvester’s identity. Chebotarev’s theorem on the matrix ř ř ε ij ř ř p−1 1 , where ε = exp(2πi/p). Problems 3. The Schur complement Given A = ţ A 11 A 12 A 21 A 22 ű , the matrix (A|A 11 ) = A 22 − A 21 A −1 11 A 12 is called the Schur complement (of A 11 in A). 3.1. det A = det A 11 det (A|A 11 ). 3.2. Theorem. (A|B) = ((A|C)|(B|C)). Problems 4. Symmetric functions, sums x k 1 +···+x k n , and Bernoulli numbers Determinant relations between σ k (x 1 , . . . , x n ), s k (x 1 , . . . , x n ) = x k 1 +···+ x k n and p k (x 1 , . . . , x n ) = P i 1 + i k =n x i 1 1 . . . x i n n . A determinant formula for S n (k) = 1 n + ··· + (k − 1) n . The Bernoulli numbers and S n (k). 4.4. Theorem. Let u = S 1 (x) and v = S 2 (x). Then for k ≥ 1 there exist polynomials p k and q k such that S 2k+1 (x) = u 2 p k (u) and S 2k (x) = vq k (u). Problems Solutions Chapter II. Linear spaces Historical remarks: Hamilton and Grassmann 5. The dual space. The orthogonal complement Linear equations and their application to the following theorem: 5.4.3. Theorem. If a rectangle with sides a and b is arbitrarily cut into squares with sides x 1 , . . . , x n then x i a ∈ Q and x i b ∈ Q for all i. Typeset by A M S-T E X 1 2 Problems 6. The kernel (null space) and the image (range) of an operator. The quotient space 6.2.1. Theorem. Ker A ∗ = (Im A) ⊥ and Im A ∗ = (Ker A) ⊥ . Fredholm’s alternative. Kronecker-Capelli’s theorem. Criteria for solv- ability of the matrix equation C = AXB. Problem 7. Bases of a vector space. Linear independence Change of basis. The characteristic polynomial. 7.2. Theorem. Let x 1 , . . . , x n and y 1 , . . . , y n be two bases, 1 ≤ k ≤ n. Then k of the vectors y 1 , . . . , y n can be interchanged with some k of the vectors x 1 , . . . , x n so that we get again two bases. 7.3. Theorem. Let T : V −→ V be a linear operator such that the vectors ξ, T ξ, . . . , T n ξ are linearly dependent for every ξ ∈ V . Then the operators I, T, . . . , T n are linearly dependent. Problems 8. The rank of a matrix The Frobenius inequality. The Sylvester inequality. 8.3. Theorem. Let U be a linear subspace of the space M n,m of n ×m matrices, and r ≤ m ≤ n. If rank X ≤ r for any X ∈ U then dim U ≤ rn. A description of subspaces U ⊂ M n,m such that dim U = nr. Problems 9. Subspaces. The Gram-Schmidt orthogonalization process Orthogonal projections. 9.5. Theorem. Let e 1 , . . . , e n be an orthogonal basis for a space V , d i = ř ř e i ř ř . The projections of the vectors e 1 , . . . , e n onto an m-dimensional subspace of V have equal lengths if and only if d 2 i (d −2 1 + ··· + d −2 n ) ≥ m for every i = 1, . . . , n. 9.6.1. Theorem. A set of k -dimensional subspaces of V is such that any two of these subspaces have a common (k − 1)-dimensional subspace. Then IVIG, PACEMAKER AND ECMO IN ACUTE MYOCARDITIS Dr Nguyen Khiet Tam Department of Cardiology Children Hospital 2- HCMC INTRODUCTION • Definition: Myocarditis is defined by an inflammation of the myocardium diagnosed by established histological, immunological, and immunohistochemical criteria • Etiology: infections (virus, bacteria, fungi, parasitic and protozoa), drugs, toxic substances, and autoimmune diseases Pediatric myocarditis: presenting clinical characteristics Yamini Durani, MD, Matthew Egan, MD, Jeanne Baffa, MD, Steven M Selbst, MD, Alan L Nager, MD http://dx.doi.org/10.1016/j.ajem.2008.07.032 Methods A retrospective cross-sectional study was conducted to identify patients with myocarditis and DCM who presented over a 10year span at tertiary care pediatric hospitals Patients were identified based on the International Classification of Diseases, Ninth Revision, diagnostic codes Results Common primary complaints were shortness of breath, vomiting, poor feeding, upper respiratory infection (URI), and fever Common examination findings were tachypnea, hepatomegaly, respiratory distress, fever, and abnormal lung examination result Sixty-three percent had cardiomegaly on chest x-ray, and all had an abnormal electrocardiogram results Conclusions These data suggest children with acute myocarditis and DCM most commonly present with difficulty breathing Myocarditis and DCM may mimic other respiratory or viral illnesses, but hepatomegaly or the finding of cardiomegaly and an abnormal electrocardiogram result may help distinguish these diagnoses from other more common pediatric illnesses Some recommendations on diagnosis Standard 12-lead ECG should be performed in all patient with clinically suspected myocarditis All patient with clinically suspected myocarditis should undergo a standard trans-thoracic echo at presentation Trans-thoracic echo should be repeated during hospitalization if there is any worsening of hemodynamics Troponin, VS, CRP should be assessed in all patient TREATMENT Hemodynamically unstable patients: cardiogenic shock, severe ventricular dysfunction  ECMO? Hemodynamically stable patients: heart failure Arrhythmia: not different from others’ causes Avoidance of exercise: physical activity should be restricted during the acute phase and for at least months IVIG IN MYOCARDITIS IVIG IN MYOCARDITIS • Insufficient data • Because of the risk of death and significant morbidity associated with myocarditis, we administer high dose IVIG (2 g/kg over 24 hours) for children with acute myocarditis, which is confirmed by endomyocardial biopsy, or when clinical suspicion is high Treatment and prognosis of myocarditis in children- Uptodate 2016 PACEMAKER IN MYOCARDITIS ECMO IN MYOCARDITIS Indication and timing of extracorporeal life support The following criteria were indications for ECLS therapy (i) cardiac and pulmonary failure refractory to medical therapy, (ii) uncontrolled arrhythmia including ventricular tachycardia or ventricular fibrillation, (iii) cardiac arrest, and (iv) low left ventricular ejection fraction (LVEF) 0,2 mcg/kg/phút dobutamin > 20 mcg/kg/phút  noradrenalin 0,2 mcg/kh/phút tối ưu tiền tải; siêu âm tim CI < 2,2 L/phút/m2; EF < 20 %, lactate máu > mmol/L (ICU BV Chợ Rẫy) THANK YOU FOR YOUR LISTENING 1 Open Hearing: Adolescent Sexual and Reproductive Health and Rights in the Pacific Questions and Answers • Why is the New Zealand Parliamentarians Group on Population and Development (NZPPD) hosting this Open Hearing? The New Zealand Parliamentarians’ Group on Population and Development (NZPPD) provides a forum for New Zealand parliamentarians to engage and act on international population and development issues. The NZPPD is a cross party group, currently with 47 members representing just under 40 percent of all New Zealand MPs. NZPPD was established in 1998 to further the International Conference on Population and Development (ICPD). The aim of this Open Hearing is to better understand adolescent sexual and reproductive health and rights and related issues in Pacific Island Countries and Territories (PICTs). Through inputs on issues and solutions from people and organisations working on the theme, NZPPD wishes to motivate increased awareness, action, and investment in adolescent SRHR in the Pacific. • What are sexual and reproductive health and rights? Sexual and reproductive health and rights (SRHR) is a phrase used to encompass fundamental human rights relating to people’s sexual and reproductive health, as well as services that are required to ensure that all people can fully realise these rights. Such services encompass, but are not limited to: family planning services, including a comprehensive range of contraceptives, commodities and information, education and counselling services on sexuality, sexual and reproductive health and responsible parenting; pre- and postnatal care services, including safe delivery services and education; infertility services; safe abortion services (where legal) and post abortion care services; treatment of reproductive tract infections and sexually transmitted infections, including HIV and other reproductive conditions. • Why is there a need to focus on adolescent SRHR in the Pacific? Firstly, adolescents have a fundamental human right to the highest attainable standard of health, including sexual and reproductive health. While studies indicate that 65% of girls and 72% of adolescents in the Pacific are sexually active, there is a severe lack of knowledge and access to comprehensive SRHR services and information. As a result, adolescents suffer a disproportionate burden of poor sexual and reproductive health, including high rates of teenage pregnancies, and sexual and reproductive illnesses. 2 Secondly, the Pacific has a rapidly growing adolescent population, with 56% of the population under the age of 25 and 36% under the age of 15. Ensuring that these adolescents have access to SRHR services and information is therefore fundamental to avoid slowed progress towards social and economic development goals, and an increased burden on government expenditures. • What is the unmet need for family planning services and information for adolescents in the Pacific? The unmet need for family planning refers to the percentage of women between the ages of 15-49 who are married or in a union, and who do not want another birth, but are not using contraception. There is a great need to increase adolescent’s access to Journal of Literature and Science 5 (2012) Challis, “Fashioning Archaeology into Art”: 53-69 53 © JLS 2012. All rights reserved. Not for unauthorised distribution. Downloaded from <http://literatureandscience.research.glam.ac.uk/journal/> Fashioning Archaeology into Art: Greek Sculpture, Dress Reform and Health in the 1880s Debbie Challis Drapery in sculpture and art has a function. It acts as clothing: as a way of both seeing and yet obscuring the figure. It draws attention to the body while covering it. It often lies next to a nude as fallen clothing. It plays a part in the narratives of sculpted story telling. It indicates how the female form should be seen and what parts of the body should be made visible through the draped veiling. Drapery has been an influential artistic conceit in the Western world since early antiquity and artists have revisited the form and function of drapery and the body since the early Renaissance. Gillian Clarke has argued that classical drapery is so prevalent in European art that “classicists tend to think of it not as clothing but as an example of Greek and Roman art” (105). Drapery has long been an ‘artistic conceit’, a device showing artistic flair and rendering. This is brought to an apogee in the large paintings by the contemporary artist Alison Watt. The contours of flesh hidden by the folds of cloth are searched for in vain as there is no body hidden. Alison Watt’s work is a study of cloth, of folds, of voids, of form for its own sake. It is what Anne Hollander has referred to as empty drapery (36), or, perhaps more positively as Gen Doy ventures, arranged cloth as art (230). The natural instinct to look for the body beneath the drapes is dictated partly by the use of drapery to show off the body, particularly in the work of nineteenth-century artists. By the end of the nineteenth century, Greek sculpture and the clothed female form was being used in an ideological and social battle – the battle for the uncorseted body. The influence of Greek sculptural ideals and Greek clothing are relatively well known, as is the connection between the aesthetic and Pre-Raphaelite artists and dress reform (Newton; Cunningham). The exhibition The Cult of Beauty. The Aesthetic Movement 1869–1900 at the Victoria and Albert Museum in 2011 made these connections through a display of clothing, dress manuals and other items. The Cult of Beauty also illustrated the influence of Greek clothing on contemporary art and dress reform through the display of two ‘Tanagra’ terracotta figurines, on loan from the British Museum, as part of the section on “Grecian Ideals.” 1 In 1879 the writer on fashion in clothing and furnishings Eliza Haweis included four sketches of these ‘Tanagra’ terracottas to illustrate how Greek women dressed using “numberless folds to both reveal and conceal the body” in her dress reform book The Art of Beauty: How gracefully the dress followed the movements of the body, may be perceived better from the small coloured clay figures in the British Museum [Greek Room], than even from marble statues, for they represent their ordinary domestic manners and are not carefully posed and idealised goddesses. (46) Dresses on display in TYPE-RAISING AND DIRECTIONALITY IN COMBINATORY GRAMMAR* Mark Steedman Computer and Information Science, University of Pennsylvania 200 South 33rd Street Philadelphia PA 19104-6389, USA (Interact: steedman@cis, upenn, edu) ABSTRACT The form of rules in ¢ombinatory categorial grammars (CCG) is constrained by three principles, called "adja- cency", "consistency" and "inheritance". These principles have been claimed elsewhere to constrain the combinatory rules of composition and type raising in such a way as to make certain linguistic universals concerning word order under coordination follow immediately. The present paper shows that the three principles have a natural expression in a unification-based interpretation of CCG in which di- rectional information is an attribute of the arguments of functions grounded in string position. The universals can thereby be derived as consequences of elementary assump- tions. Some desirable results for grammars and parsers fol- low, concerning type-raising rules. PRELIMINARIES In Categorial Grammar (CG), elements like verbs are associated with a syntactic "category", which identi- fies their functional type. I shall use a notation in which the argument or domain category always ap- pears to the right of the slash, and the result or range category to the left. A forward slash / means that the argument in question must appear on the right, while a backward slash \ means it must appear on the left. (1) enjoys := (S\NP)/NP The category (S\NP)/NP can be regarded as both a syntactic and a semantic object, in which symbols like S are abbreviations for graphs or terms including interpretations, as in the unification-based categorial grammars ofZeevat et al. [8] and others (and cf. [6]). Such functions can combine with arguments of the appropriate type and position by rules of functional application, written as follows: (2) The Functional Application Rules: a. X/Y Y =~ X (>) b. Y X\Y :=~ X (<) Such rules are also both syntactic and semantic rules *Thanks to Michael Niv and Sm Shieber. Support from: NSF Grant CISE IIP CDA 88-22719, DARPA grant no. N0014-90J- 1863, and ARO grant no. DAAL03-89-C0031. of combination in which X and Y are abbreviations for more complex objects which combine via unifi- cation. They allow context-free derivations like the following (the application of rules is indicated by in- dices >, < on the underlines: (3) Mary enjoys ~usicals m, (s\m')/~ ]w > s\lP < s The derivation can be assumed to build a composi- tional interpretation, (enjoy' musicals') mary', say. Coordination can be included in CG via the follow- ing rule, allowing constituents of like type to conjoin to yield a single constituent of the same type: (4) X conj X =~ X (5) I love and admire musicals (s\m')/m, The rest of the derivation is exactly as in (3). In order to allow coordination of contiguous strings that do not constitute constituents, CCG allows certain operations on functions related to Curry's combina- tots [1]. Functions may compose, as well as apply, under rules like the following: (6) Forward Composition: X/Y Y/Z ~B X/Z (>B) The rule corresponds to Curry's eombinator B, as the subscripted arrow indicates. It allows sentences like Mary admires, and may enjoy, musicals to be ac- cepted, via the functional composition of two verbs (indexed as >B), to yield a composite of the same category as a transitive verb. Crucially, composition also yields the appropriate interpretation for the com- posite verb may prefer in this sentence (the rest of the derivation is as in (3)): 71 (7) admires and may [...]... confused since its holders (ashabu-ha) contract the land by sale, rental, sharecropping and suchlike and pay its kharaj tax to the military or to other persons appointed by the sultan, but if they sell [their land], then the person appointed by the sultan to collect the tax takes part of the price And if they die leaving sons, then the sons take all the inheritance to the exclusion of other heirs, and no... immaterial ‘things’.2 How then are the persons and the objects in the social relationship of ownership defined in the notices above? The persons are of two kinds: institutional personae and individual persons Anthropology generally focuses on the second, but let us begin with the first Central to the making of property here are the court, the tapu office, and the governor’s office above them The court issues the basic... [Second thesis] If we say not that the lands are owned by their occupants but rather that the essence (raqaba) belongs to the treasury as has been the practice in our time and the time of our fathers and grandfathers; that when the sultan conquered a region he did not divide its land among those entitled to a part in the booty since the imam may choose between dividing the land and keeping it for the Muslims... notice of the judgment in Arabic, dated in the Ottoman financial calendar, whereas the notice sent to the tapu office is in Turkish with the date of the decision in the hijri calendar The institutional personae have defined responsibilities: the court to judge by the law on the challenge mounted against the original sale, the tapu office to amend the title registers and issue new deeds accordingly, and the governor’s... of the judgment, all the more so as the challengers to the sale were refusing to hand over the deeds they had received for the land The language of the court is formal and technical in both Ottoman Turkish and Arabic, the latter the mother tongue of the villagers The double forms of dating reflect the court’s character as part of Ottoman government as well as its august Islamic genealogy Before the institutional... negotiate the representation of rights in land in accordance with the law, the techniques of registration, the character of tax accounting, and the social organization of production on the ground Individual property rights were constructed at the intersection of law, administration and production The three parts of this book consider these in turn With regard to law, the nineteenth-century Tanzimat reforms... chapter examines doctrinal debate concerning major aspects of the land and ... complaints were shortness of breath, vomiting, poor feeding, upper respiratory infection (URI), and fever Common examination findings were tachypnea, hepatomegaly, respiratory distress, fever, and. .. acute myocarditis, which is confirmed by endomyocardial biopsy, or when clinical suspicion is high Treatment and prognosis of myocarditis in children- Uptodate 2016 PACEMAKER IN MYOCARDITIS ECMO. .. PACEMAKER IN MYOCARDITIS ECMO IN MYOCARDITIS Indication and timing of extracorporeal life support The following criteria were indications for ECLS therapy (i) cardiac and pulmonary failure refractory