Phạm Đình Đồng Exercises in Functional Analysis A review for final exam 2008 1st Edition Lời tựa To all the girls i love before. Tôi đến với giải tích hàm như một "sự sắp đặt của số phận". Có lẽ, đó là nguyên nhân để tôi việc viết tập tài liệu nhỏ này. Xin nhấn mạnh rằng, đây chỉ là sự góp nhặt khai triển chẳng có gì là sáng tạo. Thỉnh thoảng có đôi lời khen tặng, tôi lấy làm xấu hổ như đã cưỡng chiếm một cái gì đó không phải phận mình được hưởng. Khi một kẻ bình thường quên ước lượng tài sức của mình, viết về một điều quá rộng lớn và trừu tượng chắc hẳn không thể tránh khỏi thiếu sót. Rất mong sự chỉ giáo của các độc giả. Nước muôn sông không đủ cho tôi rửa tai để nghe những lời cao luận. Huế, tháng 5, 2008. Phạm Đình Đồng Ph.D.Dong 3 "A journey of a thousand miles begin with one step" - Lão Tử 1 Không gian định chuẩn Bài tập 1.1. Cho X là một không gian vectơ , f 1 , f 2 : X −→ K là các ánh xạ tuyến tính thỏa f 1 (x)f 2 (x) = 0,∀x ∈ X. Chứng minh rằng f 1 ≡ 0 hoặc f 2 ≡ 0. Chứng minh. Giả sử f 1 = 0 ta cần chứng minh f 2 = 0. Vì f 1 = 0 nên tồn tại x 1 ∈ X sao cho f 1 (x 1 ) = 0, lúc đó f 2 (x 1 f 1 (x 1 )) = f 2 (x 1 )f 1 (x 1 ) = 0 Suy ra f 2 (x 1 ) = 0 hay x 1 ∈ Kerf 2 . Nếu f 2 = 0 lúc đó tồn tại x 2 ∈ X sao cho f 2 (x 2 ) = 0 thì x 2 ∈ Kerf 1 . Đặt x 0 = x 1 + x 2 , lúc đó f 1 (x 0 ) = f 1 (x 1 ) + f 1 (x 2 ) = f 1 (x 1 ) = 0 f 2 (x 0 ) = f 2 (x 1 ) + f 2 (x 2 ) = f 2 (x 2 ) = 0 =⇒ f 1 (x 0 )f 2 (x 0 ) = f 1 (x 1 )f 2 (x 2 ) = 0 Mâu thuẫn với giả thiết, vậy f 2 ≡ 0. Bài tập 1.2. Cho X là không gian vectơ , A : X −→ X là ánh xạ tuyến tính thỏa A 2 = 0. Chứng minh rằng Id − A là song ánh. Chứng minh. Với mọi x 1 , x 2 ∈ X thỏa (Id − A)(x 1 ) = (Id − A)(x 2 ) ⇒ x 1 − A(x 1 ) = x 2 − A(x 2 ) ⇒ A(x 1 − x 2 ) = x 1 − x 2 ⇒ A 2 (x 1 − x 2 ) = A(x 1 ) − A(x 2 ) = 0 ⇒ A(x 1 ) = A(x 2 ). từ đó suy ra x 1 = x 2 . Vậy Id− A là đơn ánh. Với mọi y ∈ X, xét x = A(y)+y ∈ X, khi đó (Id−A)(x) = (Id−A)(A(y)+ y) = A(y) + y − A(A(y) + y) = A(y) + y − A 2 (y) − A(y) = y. Vậy Id − A là toàn ánh. Vậy Id − A là song ánh. Bài tập 1.3. Cho X, Y là hai không gian vectơ với dimX = n, dimY = m. Chứng minh rằng dim(L(X, Y )) = n.m. Chứng minh. Ta có L(X, Y ) = {f : X −→ Y là các ánh xạ tuyến tính } là một không gian vectơ . Lúc đó L(X, Y ) ∼ = Mat n×m (K), suy ra dim(L(X, Y )) = dimMat n×m (K). Mặt khác ta thấy A ij là ma trận sao cho a ij = 1, 1 ≤ i ≤ n, 1 ≤ j ≤ m còn các vị trí còn lại bằng 0 thì lúc đó hệ gồm {(A ij )}, 1 ≤ i ≤ n, 1 ≤ j ≤ m Ph.D.Dong 4 là độc lập tuyến tính. Mặt khác A = a 11 . . . a 1n a 21 . . . a 2n . . . . . . . . . a m1 . . . a mn thì A = n i=1 m j=1 a ij A ij Do đó {A ij } là hệ sinh của Mat n×m (K). Vậy {A ij } là cơ sở của Mat n×m (K) và nó có m × n phần tử. Vậy dim(L(X, Y )) = n.m. Bài tập 1.4. Cho f : X −→ R là ánh xạ tuyến tính , Y ⊂ X thỏa Kerf ⊂ Y . Chứng minh rằng Y = X hoặc Y = Kerf. Chứng minh. Giả sử Y là không gian con của X chứa Kerf thực sự. Lúc đó có y 0 ∈ Y và y 0 /∈ Kerf nên f(y 0 ) = 0. Với mọi x ∈ X, ta đặt z = x − f(x) f(y 0 ) y 0 thì f(z) = f(x− f(x) f(y 0 ) y 0 ) = f(x)− f(x) f(y 0 ) f(y 0 ) = f(x)− f(x) = 0 ⇒ z = x − f(x) f(y 0 ) y 0 ∈ Kerf ⊂ Y Suy ra x = z + f(x) f(y 0 ) y 0 ∈ Y , tức là X = Y . Bài tập 1.5. Cho X = {0} là không gian vectơ thực hoặc phức. Chứng minh rằng ta có thể trang bị ít nhất một chuẩn trên X. Chứng minh. Gọi B = {e α | α ∈ I} là cơ sở Hamel của X trên K. Lúc đó mọi x ∈ X, x = 0 có thể viết duy nhất dưới dạng x = n j=1 x i j e i j trong đó n ∈ N, x i j ∈ K \ {0}, i j ∈ I, j = 1, n đôi một phân biệt. Ta định nghĩa x = n j=1 x i j và x = 0 nếu x = 0 Ta sẽ chứng minh . là một chuẩn trên X. Thật vậy, Ph.D.Dong 5 • Lấy x ∈ X, x = 0. Lúc đó x = n j=1 x i j e i j trong đó n ∈ N, x i j ∈ K \ {0}, i j ∈ I, j = 1, n đôi một phân biệt. Vì x = 0 nên tồn tại ít nhất một i j = 0. Do đó, x > 0. • Với mọi x ∈ X và λ ∈ K, nếu x = 0 hoặc λ = 0 thì λx = 0, do đó λx = |λ|x. Giả sử x = 0, λ AggregateDemandinKeynesianAnalysisAggregateDemandinKeynesianAnalysis By: OpenStaxCollege The Keynesian perspective focuses on aggregatedemand The idea is simple: firms produce output only if they expect it to sell Thus, while the availability of the factors of production determines a nation’s potential GDP, the amount of goods and services actually being sold, known as real GDP, depends on how much demand exists across the economy This point is illustrated in [link] The Keynesian AD/AS Model The Keynesian View of the AD/AS Model uses an SRAS curve, which is horizontal at levels of output below potential and vertical at potential output Thus, when beginning from potential output, any decrease in AD affects only output, but not prices; any increase in AD affects only prices, not output Keynes argued that, for reasons we explain shortly, aggregatedemand is not stable—that it can change unexpectedly Suppose the economy starts where AD intersects SRAS at P0 and Yp Because Yp is potential output, the economy is at full employment Because AD is volatile, it can easily fall Thus, even if we start at Yp, if AD falls, then we find ourselves in what Keynes termed a recessionary gap The economy is in equilibrium but with less than full employment, as shown at Y1 in the [link] Keynes believed that the economy would tend to stay in a recessionary gap, with its attendant unemployment, for a significant period of time 1/7 AggregateDemandinKeynesianAnalysisIn the same way (though not shown in the figure), if AD increases, the economy could experience an inflationary gap, where demand is attempting to push the economy past potential output As a consequence, the economy experiences inflation The key policy implication for either situation is that government needs to step in and fill the gap, increasing spending during recessions and decreasing spending during booms to return aggregatedemand to match potential output Recall from The Aggregate Supply-Aggregate Demand Model that aggregatedemand is total spending, economy-wide, on domestic goods and services (Aggregate demand (AD) is actually what economists call total planned expenditure Read the appendix on The Expenditure-Output Model for more on this.) You may also remember that aggregatedemand is the sum of four components: consumption expenditure, investment expenditure, government spending, and spending on net exports (exports minus imports) In the following sections, we will examine each component through the Keynesian perspective What Determines Consumption Expenditure? Consumption expenditure is spending by households and individuals on durable goods, nondurable goods, and services Durable goods are things that last and provide value over time, such as automobiles Nondurable goods are things like groceries—once you consume them, they are gone Recall from The Macroeconomic Perspective that services are intangible things consumers buy, like healthcare or entertainment Keynes identified three factors that affect consumption: • Disposable income: For most people, the single most powerful determinant of how much they consume is how much income they have in their take-home pay, also known as disposable income, which is income after taxes • Expected future income: Consumer expectations about future income also are important in determining consumption If consumers feel optimistic about the future, they are more likely to spend and increase overall aggregatedemand News of recession and troubles in the economy will make them pull back on consumption • Wealth or credit: When households experience a rise in wealth, they may be willing to consume a higher share of their income and to save less When the U.S stock market rose dramatically in the late 1990s, for example, U.S rates of saving declined, probably in part because people felt that their wealth had increased and there was less need to save How people spend beyond their income, when they perceive their wealth increasing? The answer is borrowing On the other side, when the U.S stock market declined about 40% from March 2008 to March 2009, people felt far greater uncertainty about their economic future, so rates of saving increased while consumption declined 2/7 AggregateDemandinKeynesianAnalysis Finally, Keynes noted that a variety of other factors combine to determine how much people save and spend If household preferences about saving shift in a way that encourages consumption rather than saving, then AD will shift out to the right Visit this website for more information about how the recession affected various groups of people What Determines Investment Expenditure? Spending on new capital goods is called investment expenditure Investment falls into four categories: producer’s durable equipment and software, nonresidential structures (such as factories, offices, and retail locations), changes in inventories, and residential structures (such as single-family homes, townhouses, and apartment buildings) ... Chapter 8. VoIP: An In-Depth Analysis To create a proper network design, it is important to know all the caveats and inner workings of networking technology. This chapter explains many of the issues facing Voice over IP (VoIP) and ways in which Cisco addresses these issues. Standard time-division multiplexing (TDM) has its own set of problems, which are covered in Chapter 1, "Overview of the PSTN and Comparisons to Voice over IP," and Chapter 2, "Enterprise Telephony Today." VoIP technology has many similar issues and a whole batch of additional ones. This chapter details these various issues and explains how they can affect packet networks. The following issues are covered in this chapter: • Delay/latency • Jitter • Digital sampling • Voice compression • Echo • Packet loss • Voice activity detection • Digital-to-analog conversion • Tandem encoding • Transport protocols • Dial-plan design Delay/Latency VoIP delay or latency is characterized as the amount of time it takes for speech to exit the speaker's mouth and reach the listener's ear. Three types of delay are inherent in today's telephony networks: propagation delay, serialization delay, and handling delay. Propagation delay is caused by the speed of light in fiber or copper-based networks. Handling delay—also called processing delay—defines many different causes of delay (actual packetization, compression, and packet switching) and is caused by devices that forward the frame through the network. Serialization delay is the amount of time it takes to actually place a bit or byte onto an interface. Serialization delay is not covered in depth in this book because its influence on delay is relatively minimal. Propagation Delay Light travels through a vacuum at a speed of 186,000 miles per second, and electrons travel through copper or fiber at approximately 125,000 miles per second. A fiber network stretching halfway around the world (13,000 miles) induces a one-way delay of about 70 milliseconds (70 ms). Although this delay is almost imperceptible to the human ear, propagation delays in conjunction with handling delays can cause noticeable speech degradation. Handling Delay As mentioned previously, devices that forward the frame through the network cause handling delay. Handling delays can impact traditional phone networks, but these delays are a larger issue in packetized environments. The following paragraphs discuss the different handling delays and how they affect voice quality. In the Cisco IOS VoIP product, the Digital Signal Processor (DSP) generates a speech sample every 10 ms when using G.729. Two of these speech samples (both with 10 ms of delay) are then placed within one packet. The packet delay is, therefore, 20 ms. An initial look-ahead of 5 ms occurs when using G.729, giving an initial delay of 25 ms for the first speech frame. 120 Vendors can decide how many speech samples they want to send in one packet. Because G.729 uses 10 ms speech samples, each increase in samples per frame raises the delay by 10 ms. In fact, Cisco IOS enables users to choose how many samples to put into each frame. Cisco gave DSP much of the responsibility for framing and forming packets to keep router overhead low. The Real-Time Transport Protocol (RTP) header, for example, is placed on the frame in the DSP instead of giving the router that task. Queuing Delay A packet-based network experiences delay for other reasons. Two of these are the time necessary to move the actual packet to the output queue (packet switching) and queuing delay. When packets are held in a queue because of congestion on an outbound interface, the result is queuing delay. Queuing delay occurs when more packets are sent out than the interface can handle at a given interval. Cisco IOS software is good at moving and determining the destination of a packet. Other packet-based solutions, including PC-based solutions, are not as good Copyright 2011 Pearson Canada Inc. 24 - 1 Chapter 24 AggregateDemand and Supply Analysis Copyright 2011 Pearson Canada Inc. 24 - 2 AggregateDemand • AggregateDemand - The relationship between the quantity of aggregate output demanded and the price level when all other variables are held constant • Based on the quantity theory of money – Determined solely by the quantity of money • Based on the components parts – Consumption, investment, government spending and net exports Y AD = C + I + G + NX Copyright 2011 Pearson Canada Inc. 24 - 3 Quantity Theory of Money Approach to AggregateDemand M =quantity of money P= price level Y= aggregate real output (real income) P x Y = total nominal spending on goods and services V = the average number of time per year that a dollar is spent Multiplying both sides by M we derive the equation of exchange which relates money supply to aggregate spending M x V = P x Y Changes inaggregate spending are determined primarily by changes in the money supply M YxP V = Copyright 2011 Pearson Canada Inc. 24 - 4 Deriving the AggregateDemand Curve • Changes in the price level induce changes in the aggregate output demanded and hence movement along the AD curve (points A, B, and C in Figure 24-1) • In the quantity theory, changes in the money supply are the primary source of changes inaggregate spending and thus shifts the AD curve. Copyright 2011 Pearson Canada Inc. 24 - 5 AggregateDemand Curve Copyright 2011 Pearson Canada Inc. 24 - 6 Behaviour of Aggregate Demand’s Component Parts Y AD = C + I + G + NX The aggregatedemand curve is downward sloping because P ↓ → M/P ↑ →i ↓ → I ↑ → Y AD ↑ and P ↓ → M/P ↑ →i ↓ → E ↓ → Y AD ↑ Copyright 2011 Pearson Canada Inc. 24 - 7 Factors that Shift AggregateDemand • An increase in the money supply shifts AD to the right because it lowers interest rates and stimulates investment spending • An increase in spending from any of the components C, I, G, NX, will also shift AD to the right Copyright 2011 Pearson Canada Inc. 24 - 8 Factors That Shift the AggregateDemand Curve I Copyright 2011 Pearson Canada Inc. 24 - 9 Factors That Shift the AggregateDemand Curve II Copyright 2011 Pearson Canada Inc. 24 - 10 Aggregate Supply • Long-run aggregate supply curve (LRAS) – Determined by amount of capital and labor and the available technology – Vertical at the natural rate of output generated by the natural rate of unemployment • Short-run aggregate supply curve (SRAS) – Wages and prices are sticky – Generates an upward sloping SRAS as firms attempt to take advantage of short-run profitability when price level rises [...]... Run Aggregate Supply Short Run Aggregate Supply Factors that Shift Short Run Aggregate Supply I • Costs of production – – – – Tightness of the labor market Expected price level Wage push Change in production costs unrelated to wages (supply shocks) Factors that Shift Short Run Aggregate Supply II Equilibrium of AS and AD in the Short Run Equilibrium of AS and AD in the Long Run I Equilibrium of AS and. .. result of past high unemployment – Natural rate of unemployment shifts upward and natural rate of output falls below full employment – Expansionary policy needed to shift aggregatedemand THAI NGUYEN UNIVERITY UNIVERSITY OF AGRICULTURE AND FORESTRY PHAM MY ANH TOPIC TITLE: MATERIAL FLOW ANALYSIS OF PHOSPHOROUS AND CHEMICAL OXYGEN DEMANDIN DOMESTIC WASTEWATER AND FOOD WASTE IN SONG CONG TOWN, VIETNAM BACHELOR THESIS Study Mode : Full-time Major : Environmental Science and Management Faculty : International Training and Development Center Batch : 2011-2016 Thai Nguyen, 30/06/2015 Thai Nguyen University of Agriculture and Forestry Degree Program: Bachelor of Environmental Science and Management Student name: Pham My Anh Student ID: DTN 1153170001 “MATERIAL FLOW ANALYSIS OF PHOSPHOROUS Thesis Title: AND CHEMICAL OXYGEN DEMANDIN DOMESTIC WASTEWATER AND FOOD WASTE IN SONG CONG TOWN, VIETNAM” Supervisor: Nguyen Huu Tho, PhD Abstract: Vietnam’s fast economic growth has to a large extent been achieved on the expense of a rapid deterioration of the natural environment, including eutrophication of local water sources Proper planning is needed to move towards a sustainable wastewater management and one recognized tool for such planning is material flow analysis (MFA) This thesis uses MFA to define the current flows of phosphorus (P) and organic matter, measured as COD, in domestic wastewater and food waste in Song Cong town, Thai Nguyen province, Vietnam The aim is further to compare two different improved wastewater management scenarios with a business-as-usual scenario The methods used to find data for the MFA are literature review, interviews and a survey questionnaire The literature review presents challenges facing the wastewater sector of Vietnam and treatment techniques for wastewater and septage The wastewater sector is affected by technical difficulties such as lack of capacity and organizational challenges as a result of adjacent and overlapping authorities Contradictions and gaps in legislation, poor governance, and problems with financing are all issues that need to be addressed Although the number of wastewater treatment plants in Vietnam is increasing, ii not more than 10% of the wastewater is being treated Various techniques are tried out in Vietnam, among others constructed treatment wetlands and activated sludge techniques, such as Sequencing Batch Reactors and Anaerobic/Anoxic/Oxic processes These and other techniques are explained and compared in the literature review From the gathered data three future scenarios for Song Cong’s wastewater and food waste treatment were created along with one of the current situation The future business-as-usual scenario (BAU-2030) shows the development in Song Cong if no changes are implemented before year 2030, while the centralized scenario (CTP2030) redirects flows of wastewater to a conventional chemical/biological treatment plant The third scenario, semi-centralized (STP-2030), implements one treatment plant with enhanced biological phosphate removal (EBPR) followed by a constructed treatment wetland, and a bigger EBPR plant followed by disinfection Both of the improved scenarios also use food waste and sludge to produce biogas and digit ate that can be used as compost in agriculture The results of the MFA indicate that if nothing is done to change the current management, a 24% increase of pollutants to the Song Cong is imminent in just 15 years In addition, if one of the improved scenarios is implemented, 92% (CTP- 2030) or 90% (STP-2030) of the P will be available for reuse in agriculture, reducing the need for artificial fertilizer Further biogas is produced, which can substitute petroleum based gas for domestic purposes or be used to generate electricity Wastewater, Material flow analysis (MFA), Keywords Phosphorus (P), Chemical oxygen demand (COD) Number of pages 52 pages Date of submission 30/09/2015 Supervisor signature iii ACKNOWLEDGEMENT My bachelor thesis in Chapter 23: AggregateDemand and Supply Analysis Tp Hồ Chí Minh - 2017 Trần Thị Ngọc Hạnh Phạm Quang Thái Nguyễn Phi Điệp OUTLINE AggregateDemand and Shifts inAggregate Supply Curves Aggregate Supply and Shifts inAggregate Supply Curves Equilibrium inAggregateDemand and Supply Analysis Changes in Equilibrium: AggregateDemand Shocks Changes in Equilibrium: Aggregate Supply Shocks AD/AS Analysis of Foreign Business Cycle Episodes The Phillips Curve and the Short-Run Aggregate Supply Curve PREVIEW ● ● Aggregate demand: total amount of output demanded at different inflation rates ● ● Equilibrium occurs at the intersection of the aggregatedemand and aggregate supply curves Aggregate supply: total amount of output that firms in the economy want to sell at different inflation rates Aggregatedemand and supply analysis will enable us to explore how aggregate output and inflation are determined AggregateDemand and Shifts inAggregate Supply Curves ● The first building block of aggregate supply and demandanalysis is the aggregatedemand curve: describes the relationship between the quantity of aggregate output demanded and the inflation rate when all other variables are held constant AggregateDemand and Shifts inAggregate Supply Curves AggregateDemand 2.Planned investment 1.Consumption spending expenditure AggregateDemand Net exports 3.Government purchases AggregateDemand and Shifts inAggregate Supply Curves AggregateDemandAggregatedemand is made up of four component parts: Consumption expenditure (C): the total demand for consumer goods and services; Planned investment spending (I): the total planned spending by business firms on new machines, factories, and other capital goods, plus planned spending on new homes; Government purchases (G): spending by all levels of government (federal, state, and local) on goods and services (paper clips, computers, computer programming, missiles, government workers, and so on); Net exports (NX): the net foreign spending on domestic goods and services, equal to exports minus imports AggregateDemand and Shifts inAggregate Supply Curves AggregateDemand Y ad = C + I + G +NX (1) AggregateDemand and Shifts inAggregate Supply Curves AggregateDemand and Shifts inAggregate Supply Curves Deriving the AggregateDemand Curve • When the inflation rate rises (π ↑), the monetary authorities will raise the real interest rate (r↑) => keep inflation from spiraling out of control • Next, the resulting higher cost of financing purchases of new physical capital makes investment less profitable and causes planned investment spending to decline (I↓) • Because, Equation 1, planned investment spending is included inaggregate demand: the decline in planned ad investment spending => Y ↓ A higher inflation rate therefore leads to a lower level of the quantity of aggregate output demanded (π↑) => (Yad↓), and so the aggregatedemand curve slopes down as in Figure π ↑ => r↑ => (I↓) => (Y ad ↓) AggregateDemand and Shifts inAggregate Supply Curves Inflation Rate, π ↑ , , ↑, ,,,↑ decreases aggregatedemand and shifts the AD curve to the left AD AD2 Aggregate Output, Y Figure 1: Leftward Shift in the AggregateDemand Curve 8/1/17 10 The Friedman-Phelps Phillips Curve Analysis This prediction of the Friedman and Phelps analysis turned out to be exactly right Starting in the 1970s, after a period of very low unemployment rates, the negative relationship between unemployment and inflation, which was so visible in the 1950s and 1960s, disappeared The Friedman-Phelps Phillips Curve Analysis Given the brilliance of Friedman's and Phelps’s work, they were both awarded Nobel Prizes CONTENT THE PHILLIPS CURVE • • THE SORT-RUN AGGREGATE SUPPLY CURVE Phillips Curve Analysisin the 1960s The Friedman-Phelps Phillips Curve Analysis • The Phillips Curve after the 1960s • Okun’s Law • Deriving the Short-Run Aggregate Supply Curve The Phillips Curve after the 1960s -The modern Phillips Curve • With the sharp rise in ... Determinants of Aggregate Demand Reasons for a Decrease in Aggregate Demand Reasons for an Increase in Aggregate Demand Consumption • Rise in taxes • Fall in income • Rise in interest • Desire to... Decrease in wealth • Fall in future expected income Consumption • Decrease in taxes • Increase in income • Fall in interest rates • Desire to save less • Rise in wealth • Rise in future expected income... income Investment • Fall in expected rate of return • Rise in interest rates • Drop in business confidence Investment • Rise in expected rate of return • Drop in interest rates • Rise in business