Articles

4.9: Reduction of Order


In this section we give a method for finding the general solution of

[label{eq:5.6.1} P_0(x)y''+P_1(x)y'+P_2(x)y=F(x)]

if we know a nontrivial solution (y_1) of the complementary equation

[label{eq:5.6.2} P_0(x)y''+P_1(x)y'+P_2(x)y=0.]

The method is called reduction of order because it reduces the task of solving Equation ef{eq:5.6.1} to solving a first order equation. Unlike the method of undetermined coefficients, it does not require (P_0), (P_1), and (P_2) to be constants, or (F) to be of any special form.

By now you shoudn’t be surprised that we look for solutions of Equation ef{eq:5.6.1} in the form

[label{eq:5.6.3} y=uy_1]

where (u) is to be determined so that (y) satisfies Equation ef{eq:5.6.1}. Substituting Equation ef{eq:5.6.3} and

[egin{align*} y'&= u'y_1+uy_1' [4pt] y'' &= u''y_1+2u'y_1'+uy_1'' end{align*}]

into Equation ef{eq:5.6.1} yields

[P_0(x)(u''y_1+2u'y_1'+uy_1'')+P_1(x)(u'y_1+uy_1')+P_2(x)uy_1=F(x). onumber]

Collecting the coefficients of (u), (u'), and (u'') yields

[label{eq:5.6.4} (P_0y_1)u''+(2P_0y_1'+P_1y_1)u'+(P_0y_1''+P_1y_1'+P_2y_1) u=F.]

However, the coefficient of (u) is zero, since (y_1) satisfies Equation ef{eq:5.6.2}. Therefore Equation ef{eq:5.6.4} reduces to

[label{eq:5.6.5} Q_0(x)u''+Q_1(x)u'=F,]

with

[Q_0=P_0y_1 quad ext{and} quad Q_1=2P_0y_1'+P_1y_1. onumber]

(It isn’t worthwhile to memorize the formulas for (Q_0) and (Q_1)!) Since Equation ef{eq:5.6.5} is a linear first order equation in (u'), we can solve it for (u') by variation of parameters as in Section 1.2, integrate the solution to obtain (u), and then obtain (y) from Equation ef{eq:5.6.3}.

Example (PageIndex{1})

  1. Find the general solution of [label{eq:5.6.6} xy''-(2x+1)y'+(x+1)y=x^2,] given that (y_1=e^x) is a solution of the complementary equation [label{eq:5.6.7} xy''-(2x+1)y'+(x+1)y=0.]
  2. As a byproduct of (a), find a fundamental set of solutions of Equation ef{eq:5.6.7}.

Solution

a. If (y=ue^x), then (y'=u'e^x+ue^x) and (y''=u''e^x+2u'e^x+ue^x), so

[egin{align*} xy''-(2x+1)y'+(x+1)y&=x(u''e^x+2u'e^x+ue^x) -(2x+1)(u'e^x+ue^x)+(x+1)ue^x &=(xu''-u')e^x.end{align*}]

Therefore (y=ue^x) is a solution of Equation ef{eq:5.6.6} if and only if

[(xu''-u')e^x=x^2, onumber]

which is a first order equation in (u'). We rewrite it as

[label{eq:5.6.8} u''-{u'over x}=xe^{-x}.]

To focus on how we apply variation of parameters to this equation, we temporarily write (z=u'), so that Equation ef{eq:5.6.8} becomes

[label{eq:5.6.9} z'-{zover x}=xe^{-x}.]

We leave it to you to show (by separation of variables) that (z_1=x) is a solution of the complementary equation

[z'-{zover x}=0 onumber]

for Equation ef{eq:5.6.9}. By applying variation of parameters as in Section 1.2, we can now see that every solution of Equation ef{eq:5.6.9} is of the form

[z=vx quad ext{where} quad v'x=xe^{-x}, quad ext{so} quad v'=e^{-x} quad ext{and} quad v=-e^{-x}+C_1. onumber]

Since (u'=z=vx), (u) is a solution of Equation ef{eq:5.6.8} if and only if

[u'=vx=-xe^{-x}+C_1x. onumber]

Integrating this yields

[u=(x+1)e^{-x}+{C_1over2}x^2+C_2. onumber]

Therefore the general solution of Equation ef{eq:5.6.6} is

[label{eq:5.6.10} y=ue^x=x+1+{C_1over2}x^2e^x+C_2e^x.]

b. By letting (C_1=C_2=0) in Equation ef{eq:5.6.10}, we see that (y_{p_1}=x+1) is a solution of Equation ef{eq:5.6.6}. By letting (C_1=2) and (C_2=0), we see that (y_{p_2}=x+1+x^2e^x) is also a solution of Equation ef{eq:5.6.6}. Since the difference of two solutions of Equation ef{eq:5.6.6} is a solution of Equation ef{eq:5.6.7}, (y_2=y_{p_1}-y_{p_2}=x^2e^x) is a solution of Equation ef{eq:5.6.7}. Since (y_2/y_1) is nonconstant and we already know that (y_1=e^x) is a solution of Equation ef{eq:5.6.6}, Theorem 5.1.6 implies that ({e^x,x^2e^x}) is a fundamental set of solutions of Equation ef{eq:5.6.7}.

Although Equation ef{eq:5.6.10} is a correct form for the general solution of Equation ef{eq:5.6.6}, it is silly to leave the arbitrary coefficient of (x^2e^x) as (C_1/2) where (C_1) is an arbitrary constant. Moreover, it is sensible to make the subscripts of the coefficients of (y_1=e^x) and (y_2=x^2e^x) consistent with the subscripts of the functions themselves. Therefore we rewrite Equation ef{eq:5.6.10} as

[y=x+1+c_1e^x+c_2x^2e^x onumber]

by simply renaming the arbitrary constants. We’ll also do this in the next two examples, and in the answers to the exercises.

Example (PageIndex{2})

  1. Find the general solution of [x^2y''+xy'-y=x^2+1, onumber] given that (y_1=x) is a solution of the complementary equation [label{eq:5.6.11} x^2y''+xy'-y=0.] As a byproduct of this result, find a fundamental set of solutions of Equation ef{eq:5.6.11}.
  2. Solve the initial value problem [label{eq:5.6.12} x^2y''+xy'-y=x^2+1, quad y(1)=2,; y'(1)=-3.]

Solution

a. If (y=ux), then (y'=u'x+u) and (y''=u''x+2u'), so

[egin{aligned} x^2y''+xy'-y&=x^2(u''x+2u')+x(u'x+u)-ux &=x^3u''+3x^2u'.end{aligned}]

Therefore (y=ux) is a solution of Equation ef{eq:5.6.12} if and only if

[x^3u''+3x^2u'=x^2+1, onumber]

which is a first order equation in (u'). We rewrite it as

[label{eq:5.6.13} u''+{3over x}u'={1over x}+{1over x^3}.]

To focus on how we apply variation of parameters to this equation, we temporarily write (z=u'), so that Equation ef{eq:5.6.13} becomes

[label{eq:5.6.14} z'+{3over x}z={1over x}+{1over x^3}.]

We leave it to you to show by separation of variables that (z_1=1/x^3) is a solution of the complementary equation

[z'+{3over x}z=0 onumber]

for Equation ef{eq:5.6.14}. By variation of parameters, every solution of Equation ef{eq:5.6.14} is of the form

[z={vover x^3} quad ext{where} quad {v'over x^3}={1over x}+{1over x^3}, quad ext{so} quad v'=x^2+1 quad ext{and} quad v={x^3over 3}+x+C_1. onumber]

Since (u'=z=v/x^3), (u) is a solution of Equation ef{eq:5.6.14} if and only if

[u'={vover x^3}={1over3}+{1over x^2}+{C_1over x^3}. onumber]

Integrating this yields

[u={xover 3}-{1over x}-{C_1over2x^2}+C_2. onumber]

Therefore the general solution of Equation ef{eq:5.6.12} is

[label{eq:5.6.15} y=ux={x^2over 3}-1-{C_1over2x}+C_2x.]

Reasoning as in the solution of Example (PageIndex{1a}), we conclude that (y_1=x) and (y_2=1/x) form a fundamental set of solutions for Equation ef{eq:5.6.11}.

As we explained above, we rename the constants in Equation ef{eq:5.6.15} and rewrite it as

[label{eq:5.6.16} y={x^2over3}-1+c_1x+{c_2over x}.]

b. Differentiating Equation ef{eq:5.6.16} yields

[label{eq:5.6.17} y'={2xover 3}+c_1-{c_2over x^2}.]

Setting (x=1) in Equation ef{eq:5.6.16} and Equation ef{eq:5.6.17} and imposing the initial conditions (y(1)=2) and (y'(1)=-3) yields

[egin{aligned} c_1+c_2&= phantom{-}{8over 3} c_1-c_2&= -{11over 3}.end{aligned}]

Solving these equations yields (c_1=-1/2), (c_2=19/6). Therefore the solution of Equation ef{eq:5.6.12} is

[y={x^2over 3}-1-{xover 2}+{19over 6x}. onumber]

Using reduction of order to find the general solution of a homogeneous linear second order equation leads to a homogeneous linear first order equation in (u') that can be solved by separation of variables. The next example illustrates this.

Example (PageIndex{3})

Find the general solution and a fundamental set of solutions of

[label{eq:5.6.18} x^2y''-3xy'+3y=0,]

given that (y_1=x) is a solution.

Solution

If (y=ux) then (y'=u'x+u) and (y''=u''x+2u'), so

[egin{aligned} x^2y''-3xy'+3y&=x^2(u''x+2u')-3x(u'x+u)+3ux &=x^3u''-x^2u'.end{aligned}]

Therefore (y=ux) is a solution of Equation ef{eq:5.6.18} if and only if

[x^3u''-x^2u'=0. onumber]

Separating the variables (u') and (x) yields

[{u''over u'}={1over x}, onumber]

so

[ln|u'|=ln|x|+k,quad ext{or equivalently} quad u'=C_1x. onumber]

Therefore

[u={C_1over2}x^2+C_2, onumber]

so the general solution of Equation ef{eq:5.6.18} is

[y=ux={C_1over2}x^3+C_2x, onumber]

which we rewrite as

[y=c_1x+c_2x^3. onumber]

Therefore ({x,x^3}) is a fundamental set of solutions of Equation ef{eq:5.6.18}.


4.9: Oxidation-Reduction Reactions

The term oxidation was first used to describe reactions in which metals react with oxygen in air to produce metal oxides. When iron is exposed to air in the presence of water, for example, the iron turns to rust&mdashan iron oxide. When exposed to air, aluminum metal develops a continuous, transparent layer of aluminum oxide on its surface. In both cases, the metal acquires a positive charge by transferring electrons to the neutral oxygen atoms of an oxygen molecule. As a result, the oxygen atoms acquire a negative charge and form oxide ions (O 2&minus ). Because the metals have lost electrons to oxygen, they have been oxidized oxidation is therefore the loss of electrons. Conversely, because the oxygen atoms have gained electrons, they have been reduced, so reduction is the gain of electrons. For every oxidation, there must be an associated reduction. Therefore, these reactions are known as oxidation-reduction reactions, or "redox" reactions for short.

Any oxidation must ALWAYS be accompanied by a reduction and vice versa.

Originally, the term reduction referred to the decrease in mass observed when a metal oxide was heated with carbon monoxide, a reaction that was widely used to extract metals from their ores. When solid copper(I) oxide is heated with hydrogen, for example, its mass decreases because the formation of pure copper is accompanied by the loss of oxygen atoms as a volatile product (water vapor). The reaction is as follows:

Oxidation-reduction reactions are now defined as reactions that exhibit a change in the oxidation states of one or more elements in the reactants by a transfer of electrons, which follows the mnemonic "oxidation is loss, reduction is gain", or "oil rig". The oxidation state of each atom in a compound is the charge an atom would have if all its bonding electrons were transferred to the atom with the greater attraction for electrons. Atoms in their elemental form, such as O2 or H2, are assigned an oxidation state of zero. For example, the reaction of aluminum with oxygen to produce aluminum oxide is

Each neutral oxygen atom gains two electrons and becomes negatively charged, forming an oxide ion thus, oxygen has an oxidation state of &minus2 in the product and has been reduced. Each neutral aluminum atom loses three electrons to produce an aluminum ion with an oxidation state of +3 in the product, so aluminum has been oxidized. In the formation of Al2O3, electrons are transferred as follows (the small overset number emphasizes the oxidation state of the elements):

Equation ( ef<4.4.1>) and Equation ( ef<4.4.2>) are examples of oxidation&ndashreduction (redox) reactions. In redox reactions, there is a net transfer of electrons from one reactant to another. In any redox reaction, the total number of electrons lost must equal the total of electrons gained to preserve electrical neutrality. In Equation ( ef<4.4.3>) , for example, the total number of electrons lost by aluminum is equal to the total number gained by oxygen:

The same pattern is seen in all oxidation&ndashreduction reactions: the number of electrons lost must equal the number of electrons gained. An additional example of a redox reaction, the reaction of sodium metal with chlorine is illustrated in Figure (PageIndex<1>) .

In all oxidation&ndashreduction (redox) reactions, the number of electrons lost equals the number of electrons gained.


Sections

The sheer size of data in the modern age is not only a challenge for computer hardware but also a main bottleneck for the performance of many machine learning algorithms. The main goal of a PCA analysis is to identify patterns in data PCA aims to detect the correlation between variables. If a strong correlation between variables exists, the attempt to reduce the dimensionality only makes sense. In a nutshell, this is what PCA is all about: Finding the directions of maximum variance in high-dimensional data and project it onto a smaller dimensional subspace while retaining most of the information.

PCA Vs. LDA

Both Linear Discriminant Analysis (LDA) and PCA are linear transformation methods. PCA yields the directions (principal components) that maximize the variance of the data, whereas LDA also aims to find the directions that maximize the separation (or discrimination) between different classes, which can be useful in pattern classification problem (PCA “ignores” class labels).
In other words, PCA projects the entire dataset onto a different feature (sub)space, and LDA tries to determine a suitable feature (sub)space in order to distinguish between patterns that belong to different classes.

PCA and Dimensionality Reduction

Often, the desired goal is to reduce the dimensions of a (d)-dimensional dataset by projecting it onto a ((k))-dimensional subspace (where (k<d)) in order to increase the computational efficiency while retaining most of the information. An important question is “what is the size of (k) that represents the data ‘well’?”

Later, we will compute eigenvectors (the principal components) of a dataset and collect them in a projection matrix. Each of those eigenvectors is associated with an eigenvalue which can be interpreted as the “length” or “magnitude” of the corresponding eigenvector. If some eigenvalues have a significantly larger magnitude than others, then the reduction of the dataset via PCA onto a smaller dimensional subspace by dropping the “less informative” eigenpairs is reasonable.

A Summary of the PCA Approach

  • Standardize the data.
  • Obtain the Eigenvectors and Eigenvalues from the covariance matrix or correlation matrix, or perform Singular Value Decomposition.
  • Sort eigenvalues in descending order and choose the (k) eigenvectors that correspond to the (k) largest eigenvalues where (k) is the number of dimensions of the new feature subspace ((k le d)).
  • Construct the projection matrix (mathbf) from the selected (k) eigenvectors.
  • Transform the original dataset (mathbf) via (mathbf) to obtain a (k)-dimensional feature subspace (mathbf).

Introduction

Multifetal pregnancy reduction is defined as a first-trimester or early second-trimester procedure for reducing the total number of fetuses in a multifetal pregnancy by one or more 1. In most cases, the involved gestations will be higher-order multifetal pregnancies, defined by the presence of three or more fetuses. Throughout the document, multifetal pregnancy reduction is used to refer to reduction of a higher-order multifetal pregnancy by one or more fetuses. The special case of reduction from a twin gestation to a singleton gestation is addressed as a separate issue in the document. The ethical issues involved in multifetal pregnancy reduction are complex, and no one position reflects the variety of opinions within the membership of ACOG. The purpose of this Committee Opinion is to review the ethical considerations involved in multifetal pregnancy reduction, to analyze their role in decisions regarding multifetal pregnancy reduction, and to provide a framework that can be used by obstetrician–gynecologists in counseling patients who are considering multifetal pregnancy reduction.


4.9: Reduction of Order

The FCC granted a petition for a stay of the implementation of its new framework for the 4.9 GHz band requested by the Public Safety Spectrum Alliance (PSSA).

&ldquoThe Public Safety Spectrum Alliance (PSSA) would like to thank the Federal Communications Commission for its recent decision to stay its September 30, 2020 order removing 50 MHz of the 4.9 GHz band from exclusive public safety use,&rdquo the PSSA said in a statement. &ldquoIn particular, the PSSA commends Acting Chairwomen Jessica Rosenworcel for her leadership and support in helping preserve the 4.9 GHz spectrum for the public safety community. This spectrum is vital to public safety not only to help ensure nationwide interoperability, but to facilitate the introduction of new 5G capabilities into the public-safety communications ecosystem.&rdquo

Last September, the FCC approved new rules that would allow eligible states to lease some or all of the spectrum in the band to commercial or critical infrastructure entities. Before the approval of the rules, the band had been dedicated to public-safety use, but for many years, the FCC had looked at new ways to increase use of what it described as an underused band.

Many public-safety organizations, including the PSSA, came out against the proposal, arguing that the band should continue to be dedicated to public-safety use. The PSSA, in particular advocated for the FCC to give the band to the First Responder Network Authority (FirstNet Authority) to help serve first responders&rsquo future spectrum needs.

The PSSA, the Association of Public-Safety Communications Officials (APCO) International and the National Public Safety Telecommunications Coalition (NPSTC) all filed petitions for reconsideration of the new rules. Those petitions argued that the new leasing framework would undermine the availability and utility of spectrum in the band for public-safety operations. The petitions also described the rules as &ldquoarbitrary and capricious&rdquo because they lacked a basis in the record the FCC compiled for the proceeding.

While the changes to the FCC&rsquos rules became effective December 30. Under the rules, each state must designate a state lessor and that will not become effective until the commission receives approval from the Office of Management and Budget (OMB) under the Paperwork Reduction Act. After that approval, it will announce an effective date. The FCC has not yet announced an effective date for the final rule, so the leasing framework has not yet begun.

The FCC concluded that a stay of the 4.9 GHz leasing framework is appropriate.

&ldquoSpecifically, we believe that in light of the serious questions posed by the petitions for reconsideration, the possibility of irreparable harm to current and future public-safety users of the 4.9 GHz band and to our goal of facilitating greater use of this spectrum, the extent to which a stay will further the public interest, and the fact that no parties will be injured if a stay is granted, a stay is appropriate to permit the commission to address the issues raised in the petitions for reconsideration.&rdquo

Commissioner Brendan Carr released a dissenting statement to the order. In that statement, he expressed disappointment in the order.

&ldquoIn the 4.9 GHz band, it was clear that the status quo was not working,&rdquo Carr said. &ldquoAfter almost two decades, that 50 megahertz swath of spectrum remained woefully underutilized. So, we established a framework that could allow more intensive uses to flourish, including for public safety, by empowering local leaders to determine the best options for this spectrum based on their own circumstances. &hellip This is the spectrum equivalent of taking points off the board. While I am dissenting from today&rsquos decision, I remain hopeful that we can find a way to quickly put a beneficial framework back in place. And, I am open to working with my colleagues, the public-safety community and all other stakeholders on doing exactly that.&rdquo


Resolution

  • The storage location assigned to the delivery item must be included in MRP in order to reduce the PIR while posting the goods. E.g. the field 'SLoc MRP indicator' should be blank in the MRP 4 view of material master if the storage location is maintained for the material.
  • The 'Consumption indicator' of the planned independent requirement (PBIM-ZUVKZ) should be the same with the 'Allocation indicator' of the requirements class of sales order and its deliveries (VBBE-VPZUO or VBBS-VPZUO), and provides the consumption (values '1', '2' or '3') in order to consume the PIR. You can check these values by double clicking the PIR in MD04 or checking the tables directly.
    As to the PIR reduction while issuing the goods to customer, the table field LIPS-VPZUO for delivery can be checked, that should be the same with PBIM-ZUVKZ. Normally, the values of VPZUO in the tables VBB* and LIPS are the same.
    Regarding how the the field PBIM-ZUVKZ is filled please see SAP note 772856. The value of field VPZUO for sales order/delivery is copied from the requirement class that can be customized in the transaction OVZG and its field 'Allocation ind.'. The changes made in OVZG is only applied to the sales order/delivery newly created.
    Regarding how the requirement class is determined for sales order/delivery please see the note SAP 207942.
  • The 'Planning consumption indicator' (PBIM-VERKZ) should provide a value which can consume the customer requirements. You can display this on the detail dialog box for the planned independent requirements in the stock/requirements list (MD04) or in the item view of the independent requirements maintenance transaction codes (MD61, MD62, MD63). Any changes made to this indicator affect the consumption immediately.

Here is the screenshot of MD04 for PBIM-ZUVKZ and PBIM-VERKZ.

SAP KBA 1715491 - Planned independent requirements consumption/reduction by sales order and delivery


Redox Reactions The term oxidation was originally used to describe reactions in which an element combines with oxygen. Example: The reaction between magnesium metal and oxygen to form magnesium oxide involves the oxidation of magnesium. The term reduction comes from the Latin stem meaning "to lead back." Anything that that leads back to magnesium metal therefore involves reduction. The reaction between magnesium oxide and carbon at 2000C to form magnesium metal and carbon monoxide is an example of the reduction of magnesium oxide to magnesium metal. After electrons were discovered, chemists became convinced that oxidation-reduction reactions involved the transfer of electrons from one atom to another. From this perspective, the reaction between magnesium and oxygen is written as follows. In the course of this reaction, each magnesium atom loses two electrons to form an Mg 2+ ion. And, each O2 molecule gains four electrons to form a pair of O 2- ions. Because electrons are neither created nor destroyed in a chemical reaction, oxidation and reduction are linked. It is impossible to have one without the other, as shown in the figure below. Determine which element is oxidized and which is reduced when lithium reacts with nitrogen to form lithium nitride. 6 Li(s) + N2(g) 2 Li3N(s) Chemists eventually extended the idea of oxidation and reduction to reactions that do not formally involve the transfer of electrons. Consider the following reaction. As can be seen in the figure below, the total number of electrons in the valence shell of each atom remains constant in this reaction. What changes in this reaction is the oxidation state of these atoms. The oxidation state of carbon increases from +2 to +4, while the oxidation state of the hydrogen decreases from +1 to 0. Oxidation and reduction are therefore best defined as follows. Oxidation occurs when the oxidation number of an atom becomes larger. Reduction occurs when the oxidation number of an atom becomes smaller. Determine which atom is oxidized and which is reduced in the following reaction Sr(s) + 2 H2O(l) Sr 2+ (aq) + 2 OH - (aq) + H2(g) requires Macromedia Shockwave The terms ionic and covalent describe the extremes of a continuum of bonding. There is some covalent character in even the most ionic compounds and vice versa. It is useful to think about the compounds of the main group metals as if they contained positive and negative ions. The chemistry of magnesium oxide, for example, is easy to understand if we assume that MgO contains Mg 2+ and O 2- ions. But no compounds are 100% ionic. There is experimental evidence, for example, that the true charge on the magnesium and oxygen atoms in MgO is +1.5 and -1.5. Oxidation states provide a compromise between a powerful model of oxidation-reduction reactions based on the assumption that these compounds contain ions and our knowledge that the true charge on the ions in these compounds is not as large as this model predicts. By definition, the oxidation state of an atom is the charge that atom would carry if the compound were purely ionic. For the active metals in Groups IA and IIA, the difference between the oxidation state of the metal atom and the charge on this atom is small enough to be ignored. The main group metals in Groups IIIA and IVA, however, form compounds that have a significant amount of covalent character. It is misleading, for example, to assume that aluminum bromide contains Al 3+ and Br - ions. It actually exists as Al2Br6 molecules. This problem becomes even more severe when we turn to the chemistry of the transition metals. MnO, for example, is ionic enough to be considered a salt that contains Mn 2+ and O 2- ions. Mn2O7, on the other hand, is a covalent compound that boils at room temperature. It is therefore more useful to think about this compound as if it contained manganese in a +7 oxidation state, not Mn 7+ ions. Let's consider the role that each element plays in the reaction in which a particular element gains or loses electrons.. When magnesium reacts with oxygen, the magnesium atoms donate electrons to O2 molecules and thereby reduce the oxygen. Magnesium therefore acts as a reducing agent in this reaction. The O2 molecules, on the other hand, gain electrons from magnesium atoms and thereby oxidize the magnesium. Oxygen is therefore an oxidizing agent. Oxidizing and reducing agents therefore can be defined as follows. Oxidizing agents gain electrons. Reducing agents lose electrons. Identify the oxidizing agent and the reducing agent in the following reaction. The table below identifies the reducing agent and the oxidizing agent for some of the reactions discussed in this web page. One trend is immediately obvious: The main group metals act as reducing agents in all of their chemical reactions. Typical Reactions of Main Group Metals Reaction Reducing Agent Oxidizing Agent 2 Na + Cl2 2 NaCl Na Cl2 2 K + H2 2 KH K H2 4 Li + O2 2 Li2O Li O2 2 Na + O2 Na2O2 Na O2 2 Na + 2 H2O 2 Na + + 2 OH - + H2 Na H2O 2 K + 2 NH3 2 K + + 2 NH2 - + H2 K NH3 2 Mg + O2 2 MgO Mg O2 3 Mg + N2 Mg3N2 Mg N2 Ca + 2 H2O Ca 2+ + 2 OH - + H2 Ca H2O 2 Al + 3 Br2 Al2Br6 Al Br2 Mg + 2 H + Mg 2+ + H2 Mg H + Mg + H2O MgO + H2 Mg H2O Metals act as reducing agents in their chemical reactions. When copper is heated over a flame, for example, the surface slowly turns black as the copper metal reduces oxygen in the atmosphere to form copper(II) oxide. If we turn off the flame, and blow H2 gas over the hot metal surface, the black CuO that formed on the surface of the metal is slowly converted back to copper metal. In the course of this reaction, CuO is reduced to copper metal. Thus, H2 is the reducing agent in this reaction, and CuO acts as an oxidizing agent. An important feature of oxidation-reduction reactions can be recognized by examining what happens to the copper in this pair of reactions. The first reaction converts copper metal into CuO, thereby transforming a reducing agent (Cu) into an oxidizing agent (CuO). The second reaction converts an oxidizing agent (CuO) into a reducing agent (Cu). Every reducing agent is therefore linked, or coupled, to a conjugate oxidizing agent, and vice versa. Every time a reducing agent loses electrons, it forms an oxidizing agent that could gain electrons if the reaction were reversed. Conversely, every time an oxidizing agent gains electrons, it forms a reducing agent that could lose electrons if the reaction went in the opposite direction. The idea that oxidizing agents and reducing agents are linked, or coupled, is why they are called conjugate oxidizing agents and reducing agents. Conjugate comes from the Latin stem meaning "to join together." It is therefore used to describe things that are linked or coupled, such as oxidizing agents and reducing agents. The main group metals are all reducing agents. They tend to be "strong" reducing agents. The active metals in Group IA, for example, give up electrons better than any other elements in the periodic table. The fact that an active metal such as sodium is a strong reducing agent should tell us something about the relative strength of the Na + ion as an oxidizing agent. If sodium metal is relatively good at giving up electrons, Na + ions must be unusually bad at picking up electrons. If Na is a strong reducing agent, the Na + ion must be a weak oxidizing agent. Conversely, if O2 has such a high affinity for electrons that it is unusually good at accepting them from other elements, it should be able to hang onto these electrons once it picks them up. In other words, if O2 is a strong oxidizing agent, then the O 2- ion must be a weak reducing agent. In general, the relationship between conjugate oxidizing and reducing agents can be described as follows. Every strong reducing agent (such as Na) has a weak conjugate oxidizing agent (such as the Na + ion). Every strong oxidizing agent (such as O2) has a weak conjugate reducing agent (such as the O 2- ion). We can determine the relative strengths of a pair of metals as reducing agents by determining whether a reaction occurs when one of these metals is mixed with a salt of the other. Consider the relative strength of iron and aluminum, for example. Nothing happens when we mix powdered aluminum metal with iron(III) oxide. If we place this mixture in a crucible, however, and get the reaction started by applying a little heat, a vigorous reaction takes place to give aluminum oxide and molten iron metal. By assigning oxidation numbers, we can pick out the oxidation and reduction halves of the reaction. Aluminum is oxidized to Al2O3 in this reaction, which means that Fe2O3 must be the oxidizing agent. Conversely, Fe2O3 is reduced to iron metal, which means that aluminum must be the reducing agent. Because a reducing agent is always transformed into its conjugate oxidizing agent in an oxidation-reduction reaction, the products of this reaction include a new oxidizing agent (Al2O3) and a new reducing agent (Fe). Since the reaction proceeds in this direction, it seems reasonable to assume that the starting materials contain the stronger reducing agent and the stronger oxidizing agent. In other words, if aluminum reduces Fe2O3 to form Al2O3 and iron metal, aluminum must be a stronger reducing agent than iron. We can conclude from the fact that aluminum cannot reduce sodium chloride to form sodium metal that the starting materials in this reaction are the weaker oxidizing agent and the weaker reducing agent. We can test this hypothesis by asking: What happens when we try to run the reaction in the opposite direction? (Is sodium metal strong enough to reduce a salt of aluminum to aluminum metal?) When this reaction is run, we find that sodium metal can, in fact, reduce aluminum chloride to aluminum metal and sodium chloride when the reaction is run at temperatures hot enough to melt the reactants. 3 Na(l) + AlCl3(l) 3 NaCl(l) + Al(l) If sodium is strong enough to reduce Al 3+ salts to aluminum metal and aluminum is strong enough to reduce Fe 3+ salts to iron metal, the relative strengths of these reducing agents can be summarized as follows. Use the following equations to determine the relative strengths of sodium, magnesium, aluminum, and calcium metal as reducing agents. Bill to Soften ‘Windfall’ Reduction Reintroduced

Legislation has been reintroduced in Congress (HR-2337) to soften the impact of the “windfall elimination provision” that reduces Social Security benefits of those also drawing an annuity from a retirement program that does not include Social Security, including the CSRS system.

That provision reduces the Social Security benefits earned by those persons—typically before or after federal service, or as side income while federally employed—if they have fewer than 30 years of “substantial” earnings under Social Security — this year, $26,550.

The maximum reduction of the Social Security benefit works out to be about $500 a month this year there is a lesser reduction for those with between 20 and 30 years of such earnings.

Under the bill, those already retired would receive an increase in benefits of the greater of $150 or the reduction they are subject to each month, starting nine months after enactment. Those retiring in 2023 or later would receive the greater of their current benefit or the benefit calculated under a new formula that sponsors say would be worth about $75 more a month to most.

Numerous bills have been offered over the years to either eliminate or soften the windfall provision, which affects some 2 million retirees, including some retirees of state and local governments. None have come close to enactment although the current measure does have the important sponsorship of the head of the House Ways and Means Committee, Rep. Richard Neal, D-Mass.

The NARFE organization said that “while this bill does not provide WEP-affected individuals the full repeal they are due, it represents a good first step in allowing some relief from this unreasonable penalty.”

The measure would not affect another Social Security reduction applying to CSRS system retirees, the “government pension offset” which reduces and in many cases eliminates a spousal or survivor Social Security benefit.


The explicit methods are those where the matrix [ a i j ] ]> is lower triangular.

Forward Euler Edit

The Euler method is first order. The lack of stability and accuracy limits its popularity mainly to use as a simple introductory example of a numeric solution method.

Explicit midpoint method Edit

The (explicit) midpoint method is a second-order method with two stages (see also the implicit midpoint method below):

Heun's method Edit

Heun's method is a second-order method with two stages. It is also known as the explicit trapezoid rule, improved Euler's method, or modified Euler's method. (Note: The "eu" is pronounced the same way as in "Euler", so "Heun" rhymes with "coin"):

Ralston's method Edit

Ralston's method is a second-order method [1] with two stages and a minimum local error bound:

Generic second-order method Edit

Kutta's third-order method Edit

Generic third-order method Edit

See Sanderse and Veldman (2019). [2]

Heun's third-order method Edit

Ralston's third-order method Edit

Ralston's third-order method [3] is used in the embedded Bogacki–Shampine method.

Third-order Strong Stability Preserving Runge-Kutta (SSPRK3) Edit

Classic fourth-order method Edit

The "original" Runge–Kutta method.

Ralston's fourth-order method Edit

This fourth order method [4] has minimum truncation error.

3/8-rule fourth-order method Edit

This method doesn't have as much notoriety as the "classical" method, but is just as classical because it was proposed in the same paper (Kutta, 1901).

The embedded methods are designed to produce an estimate of the local truncation error of a single Runge–Kutta step, and as result, allow to control the error with adaptive stepsize. This is done by having two methods in the tableau, one with order p and one with order p-1.

The lower-order step is given by

Heun–Euler Edit

The simplest adaptive Runge–Kutta method involves combining Heun's method, which is order 2, with the Euler method, which is order 1. Its extended Butcher Tableau is:

The error estimate is used to control the stepsize.

Fehlberg RK1(2) Edit

The Fehlberg method [5] has two methods of orders 1 and 2. Its extended Butcher Tableau is:

0
1/2 1/2
1 1/256 255/256
1/512 255/256 1/512
1/256 255/256 0

The first row of b coefficients gives the second-order accurate solution, and the second row has order one.

Bogacki–Shampine Edit

The Bogacki–Shampine method has two methods of orders 3 and 2. Its extended Butcher Tableau is:

0
1/2 1/2
3/4 0 3/4
1 2/9 1/3 4/9
2/9 1/3 4/9 0
7/24 1/4 1/3 1/8

The first row of b coefficients gives the third-order accurate solution, and the second row has order two.

Fehlberg Edit

The Runge–Kutta–Fehlberg method has two methods of orders 5 and 4. Its extended Butcher Tableau is:

The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four.

Cash-Karp Edit

Cash and Karp have modified Fehlberg's original idea. The extended tableau for the Cash–Karp method is

0
1/5 1/5
3/10 3/40 9/40
3/5 3/10 −9/10 6/5
1 −11/54 5/2 −70/27 35/27
7/8 1631/55296 175/512 575/13824 44275/110592 253/4096
37/378 0 250/621 125/594 0 512/1771
2825/27648 0 18575/48384 13525/55296 277/14336 1/4

The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four.

Dormand–Prince Edit

The extended tableau for the Dormand–Prince method is

0
1/5 1/5
3/10 3/40 9/40
4/5 44/45 −56/15 32/9
8/9 19372/6561 −25360/2187 64448/6561 −212/729
1 9017/3168 −355/33 46732/5247 49/176 −5103/18656
1 35/384 0 500/1113 125/192 −2187/6784 11/84
35/384 0 500/1113 125/192 −2187/6784 11/84 0
5179/57600 0 7571/16695 393/640 −92097/339200 187/2100 1/40

The first row of b coefficients gives the fifth-order accurate solution and the second row gives the fourth-order accurate solution.

Backward Euler Edit

The backward Euler method is first order. Unconditionally stable and non-oscillatory for linear diffusion problems.

Implicit midpoint Edit

The implicit midpoint method is of second order. It is the simplest method in the class of collocation methods known as the Gauss-Legendre methods. It is a symplectic integrator.

Crank-Nicolson method Edit

The Crank–Nicolson method corresponds to the implicit trapezoidal rule and is a second-order accurate and A-stable method.

Gauss–Legendre methods Edit

These methods are based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method of order four has Butcher tableau:

The Gauss–Legendre method of order six has Butcher tableau:

Diagonally Implicit Runge–Kutta methods Edit

Diagonally Implicit Runge–Kutta (DIRK) formulae have been widely used for the numerical solution of stiff initial value problems. The simplest method from this class is the order 2 implicit midpoint method.

Kraaijevanger and Spijker's two-stage Diagonally Implicit Runge–Kutta method:

Qin and Zhang's two-stage, 2nd order, symplectic Diagonally Implicit Runge–Kutta method:

Pareschi and Russo's two-stage 2nd order Diagonally Implicit Runge–Kutta method:

Two-stage 2nd order Diagonally Implicit Runge–Kutta method:

Crouzeix's two-stage, 3rd order Diagonally Implicit Runge–Kutta method:

Three-stage, 3rd order, L-stable Diagonally Implicit Runge–Kutta method:

Nørsett's three-stage, 4th order Diagonally Implicit Runge–Kutta method has the following Butcher tableau:

Four-stage, 3rd order, L-stable Diagonally Implicit Runge–Kutta method

Lobatto methods Edit

There are three main families of Lobatto methods, called IIIA, IIIB and IIIC (in classical mathematical literature, the symbols I and II are reserved for two types of Radau methods). These are named after Rehuel Lobatto. All are implicit methods, have order 2s − 2 and they all have c1 = 0 and cs = 1. Unlike any explicit method, it's possible for these methods to have the order greater than the number of stages. Lobatto lived before the classic fourth-order method was popularized by Runge and Kutta.

Lobatto IIIA methods Edit

The Lobatto IIIA methods are collocation methods. The second-order method is known as the trapezoidal rule:

The fourth-order method is given by

This methods are A-stable, but not L-stable and B-stable.

Lobatto IIIB methods Edit

The Lobatto IIIB methods are not collocation methods, but they can be viewed as discontinuous collocation methods (Hairer, Lubich & Wanner 2006, §II.1.4). The second-order method is given by

The fourth-order method is given by

Lobatto IIIB methods are A-stable, but not L-stable and B-stable.

Lobatto IIIC methods Edit

The Lobatto IIIC methods also are discontinuous collocation methods. The second-order method is given by

The fourth-order method is given by

They are L-stable. They are also algebraically stable and thus B-stable, that makes them suitable for stiff problems.

Lobatto IIIC* methods Edit

The Lobatto IIIC* methods are also known as Lobatto III methods (Butcher, 2008), Butcher's Lobatto methods (Hairer et al., 1993), and Lobatto IIIC methods (Sun, 2000) in the literature. [6] The second-order method is given by

Butcher's three-stage, fourth-order method is given by

These methods are not A-stable, B-stable or L-stable. The Lobatto IIIC* method for s = 2 is sometimes called the explicit trapezoidal rule.

Generalized Lobatto methods Edit

One can consider a very general family of methods with three real parameters ( α A , α B , α C ) ,alpha _)> by considering Lobatto coefficients of the form

For example, Lobatto IIID family introduced in (Nørsett and Wanner, 1981), also called Lobatto IIINW, are given by

Radau methods Edit

Radau methods are fully implicit methods (matrix A of such methods can have any structure). Radau methods attain order 2s − 1 for s stages. Radau methods are A-stable, but expensive to implement. Also they can suffer from order reduction. The first order Radau method is similar to backward Euler method.


Percent Off Calculator

A percent off of a product or service is a common discount format. A percent off of a product means that the price of the product is reduced by that percent. For example, given a product that costs $279, 20% off of that product would mean subtracting 20% of the original price, from the original price. For example:

20% of $279 = 0.20 × 279 = $55.80

You would therefore be saving $55.80 on the purchase for a final price of $223.20.

For this calculator a "stackable additional discount" means getting a further percent off of a product, after a discount is applied. Using the same example, assume that the 20% discount is a discount applied by the store to the product. If you have a coupon for another 15% off, the 15% off would then be applied to the discounted price of $223.20. It is not a total of 35% off of the original price, it is less:

Thus, with a 20% discount off of $279, and an additional 15% off of that discounted price, you would end up saving a total of:

This equates to a 32% discount, rather than a 35% discount, and this calculation is how the calculator is intended to be used. As an example, to more efficiently compute the discount described above:

Final price = (0.80 × 279) × 0.85 = $189.72

This is because 80% of the original price is the same as subtracting 20% of the original price, from the original price. The same is true for 85% and 15% case applied to the discounted price.


Watch the video: Μείωση επιτοκίων (September 2021).