Theorem of bounded convergence. Uniformity of convergence is a sufficient condition for continuity or integrability of the sum, provided the separate terms are continuous or integrable. It is far from a necessary condition. In practice it is usually easier to test directly whether the limit function is integrable than to test for uniform convergence, and there are so many cases where the passage to the limit under the integral sign is valid without convergence being uniform that a more general rule is needed. Such a rule is as follows. It is known as the theorem of bounded convergencea. If for all $a \leqslant x \leqslant b,\left|f_n(x)\right| \leqslant M$ for all $n$ and $x$, if all $f_n(x)$ are integrable and if $f_n(x) \rightarrow f(x)$, where $f(x)$ is integrable, then $\int_a^b f_n(x) d x \rightarrow \int_a^b f(x) d x$. The proof is not easy, but the result should be known. The behaviour of $f_n(x)$ and $$ \lim {n \rightarrow \infty} \int_0^1 f_n(x) d x, \int_0^1 \lim {n \rightarrow \infty} f_n(x) d x $$ should be studied for the cases $$ f_n(x)=x e^{-n x}, f_n(x)=n x e^{-n x}, f_n(x)=n^2 x e^{-n x} $$
数学代写|数学物理方法代写Mathematical Methods代考|Useful comparison series
1.117. Useful comparison series. By far the most important comparison series are $\Sigma x^n(0 \leqslant x<1), \Sigma n^{-s}(s>1)$, which we have already studied, and $\Sigma n^s a^n(0 \leqslant a<1)$. The convergence of the latter follows at once from the $M$ test if $s<0$. If $s \geqslant 0$, we have $$ \frac{u_{n+1}}{u_n}=\left(\frac{n+1}{n}\right)^s a=\left(1+\frac{1}{n}\right)^{\circ} a \text {. } $$ As $n$ increases this tends to $a$. Hence, since $a<1$, we can take $m$ large enough for this ratio to be less than $b$ for all $n>m$, where $am$, $$ \left|u_n\right|<u_m b^{n-m} $$ and $\Sigma b^{n-m}$ is a convergent series of positive terms. Hence $\Sigma n^s a^n$ converges for $0 \leqslant a<1$. Comparison with the series $\Sigma n^{-s}$ can often be simplified. If $v_n=n^{-s}(1<8)$, $$ n\left(1-\frac{v_n}{v_{n-1}}\right)=n\left{1-\left(\frac{n-1}{n}\right)^s\right} \rightarrow 8 $$
有界收敛定理。收敛的均匀性是和的连续性或可积性的充分条件，前提是单独的项是连续的或可积的。这远非 必要条件。在实践中，直接检验极限函数是否可积通常比检验一致收敛更容易，并且有很多情况下通过积分符 号到达极限是有效的而不是一致收敛的，所以更一般的规则是需要。这样的规则如下。它被称为有界收敛定理 哪里 $f(x)$ 是可积的，那么 $\int_a^b f_n(x) d x \rightarrow \int_a^b f(x) d x$. 证明并不容易，但结果应该是已知的。的行为 $f_n(x)$ 和 $$ \lim n \rightarrow \infty \int_0^1 f_n(x) d x, \int_0^1 \lim n \rightarrow \infty f_n(x) d x $$ 应该研究案例 $$ f_n(x)=x e^{-n x}, f_n(x)=n x e^{-n x}, f_n(x)=n^2 x e^{-n x} $$
数学代写|数学物理方法代写Mathematical Methods代考|Useful comparison series
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
数学代写|数学物理方法代写Mathematical Methods代考|MST224 Infinite and improper integrals
数学代写|数学物理方法代写Mathematical Methods代考|Infinite and improper integrals
1-104. Infinite and improper integrals. The proof of the existence of an integral breaks down if either the interval $b-a$ is infinite or the function to be integrated is unbounded in the interval. In the former case, $b-a$ is infinite and we cannot make $\omega>0,(b-a) \omega<\epsilon$ by any choice of $\omega$. In the latter the approximating sum may vary to any extent according to the point chosen to sample $f(x)$ in the subinterval where $f(x)$ is unbounded. A special device is needed in either case to give a meaning to the integral. The method used for integrals with an infinite upper limit is to use first an integral with a finite upper limit; if the integral tends to a definite limit when the upper limit tends to infinity this limit is taken as the value of the infinite integral. The need for such a device may be illustrated by the integral $$ I=\int_0^{\infty} \frac{\sin x}{x} d x $$ According to our rule this must be interpreted as $$ \lim _{x \rightarrow \infty} \int_0^x \frac{\sin x}{x} d x $$
数学代写|数学物理方法代写Mathematical Methods代考|Since an integral is a function of its upper terminus
1-1041. Since an integral is a function of its upper terminus, we can adapt the tests for convergence of a sequence given in $1.0441$ and $1.045$ on the lines already mentioned in the theory of continuity (1.06). The proofs are straightforward.
If $f(x) \geqslant 0$ and $\int_a^X f(x) d x$ is bounded for all $X>a$, then $\int_a^X f(x) d x$ tends to a limit as $X \rightarrow \infty$. A necessary and sufficient condition that $\int_a^X f(x) d x$ shall tend to a limit as $X \rightarrow \infty$ is that for any positive $\epsilon$, however small, there is an $A$ such that $\left|\int_{\Delta}^X f(x) d x\right|<\epsilon$ for all $X>A$. 1-1042. The relation between infinite integrals and series is so close that the same words are convenient to express the properties: $\int_a^{\infty} f(x) d x$ is convergent if $\lim _{X \rightarrow \infty} \int_a^x f(x) d x$ exists. $\int_a^{\infty} f(x) d x$ is unbounded if $\left|\int_a^X f(x) d x\right|$ is unbounded as $X \rightarrow \infty$. $\int_a^{\infty} f(x) d x=\infty$ if $\int_a^x f(x) d x \rightarrow \infty$ as $x \rightarrow \infty$. $\int_a^{\infty} f(x) d x$ is finitely oscillatory if there are positive quantities $\omega, M$ such that for any $X$ we can choose $Y_1>X$ so that $\left|\int_X^{Y_1} f(x) d x\right|>\omega$, but cannot choose $Y_2$ so that $\left|\int_X^{Y_1} f(x) d x\right|>M$.
数学代写|数学物理方法代写Mathematical Methods代考|MST224 Infinite and improper integrals
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
数学代写|数学物理方法代写Mathematical Methods代考|PCS622 Orders of magnitude
数学代写|数学物理方法代写Mathematical Methods代考|Orders of magnitude
Orders of magnitude. If as $x$ tends to a limit $\phi(x)$ tends to 0 or $\infty$, and $f(x) / \phi(x)$ is bounded, we say that $f(x)=O{\phi(x)}$, or that $f(x)$ is of the same order of magnitude as $\phi(x)$. If $f(x) / \phi(x) \rightarrow 0$ as $\phi(x) \rightarrow 0$ we write $f(x)=o{\phi(x)}$. If $f(x)$ is bounded we can write $f(x)=O(1)$. This notation must be distinguished from the common usage in physics, where we may say that the masses of Jupiter and Saturn are of the same order of magnitude, meaning roughly that they differ by not more than a factor 10 without there being any question of a limit. In the physical sense the quantities compared must have the same dimensions. This is not necessary in the mathematical sense. $x$ may, for instance, be a time-interval and $f(x)$ the distance travelled by a sound wave. Then $f(x)=O(x)$ because $f(x) / x$ is the velocity of sound and is supposed finite. Note that $$ O\left(x^m\right) O\left(x^n\right)=O\left(x^{m+n}\right), \quad o\left(x^m\right) O\left(x^n\right)=o\left(x^{m+n}\right) . $$
数学代写|数学物理方法代写Mathematical Methods代考|Functions of bounded variation
Functions of bounded variation. If the function $f(x)$ is defined in the closed interval $(a, b)$, and there is a number $M$ such that $$ v=\left|f\left(x_1\right)-f\left(x_0\right)\right|+\left|f\left(x_2\right)-f\left(x_1\right)\right|+\ldots+\left|f\left(x_n\right)-f\left(x_{n-1}\right)\right| \leqslant M, $$ for every subdivision $a=x_0<x_1<x_2<\ldots<x_{n-1}<x_n=b, f(x)$ is said to be of bounded variation in the interval $(a, b)$; and the upper bound of the sums $v$ for all possible selections of the subdivisions is called the total variation of $f(x)$ in the interval.* The total variation is of interest since it is related to the condition for existence of a Stieltjes integral (1.102), and to the existence of the length of a curve, and it is useful in the theory of Fourier series and Fourier integrals.
We assume repeatedly that the sum and product of two continuous functions (and therefore of any finite number) are continuous, and that those of two functions of bounded variation are of bounded variation. The proofs are simple: for the last, notice that $$ \begin{aligned} f\left(x_{r+1}\right) g\left(x_{r+1}\right)-f\left(x_r\right) g\left(x_r\right)=\frac{1}{2}\left{f\left(x_r\right)+f\left(x_{r+1}\right)\right} & \left{g\left(x_{r+1}\right)-g\left(x_r\right)\right} \ & +\frac{1}{2}\left{g\left(x_r\right)+g\left(x_{r+1}\right)\right}\left{f\left(x_{r+1}\right)-f\left(x_r\right)\right} \end{aligned} $$ and it follows that if $M, N$ are the upper bounds of $|f(x)|,|g(x)|$, and $U, V$ the total variations of $f(x), g(x)$ in the interval, the total variation of $f(x) g(x)$ is not greater than $M V+N U$.
数学代写|数学物理方法代写Mathematical Methods代考|PCS622 Orders of magnitude
数学代写|数学物理方法代写Mathematical Methods代考|Orders of magnitude
数量级。如果 $\$ x$ 趋于极限 $\$ \backslash p h i(x) \$$ 趋于 0 或 $\$ \backslash$ infty $\$$ ，并且 $\$ f(x) / \backslash p h i(x) \$$ 是有界的，我们说 $\$ f(x)=0{\backslash p h i(x)}$ ， 或者说 $\$ f(x)$ 与 $\$ \backslash p h i(x) \$$ 的数量级相同。如果 $\$ f(x) / \backslash p h i(x)$ \rightarrow $0 \$$ 为 $\$ \backslash p h i(x) \backslash r i g h t a r r o w 0 \$$ 我们写 $\$ f(x)=o{\backslash p h i(x)}$ 。如果 $\$ f(x) \$$ 是有界的，我们可以写成 $\$ f(x)=O(1) \$$ 。这种表示法必须与物理学中的常用用法区分开来， 在物理学中我们可以说木星和土星的质量具有相同的数荲级，大致意思是它们相差不超过 10 倍，而不存在任何质量问题。限制。 在物理意义上，所比较的量必须具有相同的量纲。这在数学意义上是不必要的。例如， $\$ \mathrm{x} \$$ 可以是时间间隔， $\$ \mathrm{f}(\mathrm{x}) \$$ 可以是声波 传播的距新。 注意 $\$ \$$ $O \backslash$ left $(x \wedge m \backslash$ right $)$ o left $(x \wedge n \backslash$ right $)=0 \backslash$ left $(x \wedge{m+n} \backslash$ right $)$, \quad o left $(x \wedge m \backslash$ 右 $)$ $O \backslash$ left $(x \wedge n \backslash$ right $)=o \backslash$ left $(x \wedge{m+n} \backslash$ right $)$ 。 $\$ \$$
数学代写|数学物理方法代写Mathematical Methods代考|Functions of bounded variation
有限变化的功能。如果函数 $\$ f(x) \$$ 定义在闭区间 $\$(a, b) \$$ 中，并且存在一个数 $\$ M \$$ 使得 $\$ \$$ $\$ \$$ 对于每个细分 $\$ a=x _0<x _l<x _2<\backslash l d o t s<x _{n-1}<x _n=b ， f(x) \$$ 被称为区间 $\$(a, b) \$$ 中的有界变化; 并且所有可能的细分 选择的总和 $\$ v \$$ 的上限称为区间内 $\$ f(x) \$$ 的总变差。总变差很有趣，因为它与存在的条件有关一个 Stieltjes 积分 (1.102)，以 及曲线长度的存在性，它在傅里叶级数和傅里叶积分理论中很有用。 我们反㵊假设两个连续函数（因此任何有限数）的和与积是连㩺的，并且两个有界变差函数的和与积是有界变差。证明很简单：最 后，注意 $\$ \$$ lbegin ${$ aligned $}$ $f \backslash$ left $\left(x_{-}{r+1} \backslash\right.$ right $) g \backslash$ left $\left(x_{-}{r+1} \backslash\right.$ right $)-f \backslash$ left $\left(x_{-} r \backslash r i g h t\right) g \backslash$ left $\left(x_{-} r \backslash\right.$ right $)=\backslash$ frac ${1}$ ${2} \backslash$ left $\left{f \backslash\right.$ left $\left(x_{-} r \backslash\right.$ right $)+f \backslash$ left $\left(x_{-}{r+1} \backslash r i g h t\right) \backslash$ right $} \& \backslash$ 左 $\left{g \backslash\right.$ left $\left(x_{-}{r+1} \backslash r i g h t\right)-g \backslash$ left $\left.\left(x_{-} r \backslash r i g h t\right) \backslash r i g h t\right} \backslash \&$ $+\backslash$ frac ${1}{2} \backslash$ left $\left{g \backslash\right.$ left $\left(x_{-} r \backslash\right.$ right $)+g \backslash$ left $\left(x_{-}{r+1} \backslash\right.$ right $) \backslash$ right $} \backslash$ left $\left{f \backslash l e f t\left(x_{-}{r+1} \backslash\right.\right.$ right $)-$ $f \backslash$ left $\left.\left(x _r \backslash r i g h t\right) \backslash r i g h t\right} \backslash$ lend ${$ aligned } $\$ \$$ 并且它菖砕如果 $\$ M, N \$$ 是 $\$|f(x)|,|g(x)| \$$ 的上限，并且 $\$ U, V \$$ 是 $\$ f(x)$ 的总变化， $g(x) \$ 区 i ⿵ 冂 人 ， \$ f(x) g(x) \$$ 的总变差不大 于 $\$ M V+N U \$$ 。
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
数学代写|数学物理方法代写Mathematical Methods代考|PCS622 Extension of the M test
数学代写|数学物理方法代写Mathematical Methods代考|Extension of the M test
1-1152. Extension of the $M$ test. A modification of the $M$ test is sometimes useful even for conditionally convergent series where we cannot find a convergent series of positive terms $v_n$ numerically greater than $u_n(x)$. Suppose that as $n \rightarrow \infty, u_n(x)$ tends uniformly to 0 (see below); that the terms of $\Sigma u_n(x)$ can be taken in batches of $m$ without deranging the order, giving a series $\Sigma U_\nu(x)$; and that $\left|U_\nu(x)\right|<V_\nu$, where $\Sigma V_\nu$ is convergent. Then $\Sigma U_\nu(x)$ is uniformly convergent by the $M$ test. It remains to show that in the conditions stated $\Sigma u_n(x)$ exists and is equal to $\Sigma U_\nu(x)$.
Since $u_n(x)$ tends uniformly to 0 , for any $\epsilon$ we can choose $n$ so that $\left|u_p(x)\right|<\epsilon$ for all $p \geqslant n$ and for all $x$ in the range. Then if we take $\nu$ for given $n$ so that $$ \begin{aligned} \nu m \leqslant n & <(\nu+1) m, \ \left|\sum_1^n u_n(x)-\sum_1^\nu U_\nu(x)\right| & =\left|u_{m \nu+1}(x)+\ldots+u_n(x)\right| . \end{aligned} $$ Take $\nu$ so that $\sum_{\nu+1}^{\infty} U_\sigma(x)<\frac{1}{2} \epsilon$, and so that all $\left|u_p(x)\right|<\frac{\epsilon}{2 m}$ for $p>m \nu$. Then $$ \left|\sum_1^n u_n(x)-\sum_1^{\infty} U_\nu(x)\right|<\epsilon $$ for all $x$, and all $n>m v$, and $\Sigma u_n(x)$ is uniformly convergent. This can be applied to the series $$ \sum_1^{\infty}(-1)^n \frac{n}{n^2+x^2} . $$
数学代写|数学物理方法代写Mathematical Methods代考|Abel’s lemma
1.1153. Abel’s lemma. Though the $M$ test is the commonest in actual applications, series may be uniformly convergent and not satisfy it. Two more sensitive tests are based on Abel’s lemma. All these tests have analogues for integrals. If $\left{v_r\right}$ is a non-increasing sequence of non-negative quantities, and if the sums $$ s_p=a_1+a_2+\ldots a_p $$ satisfy the inequalities $h \leqslant \delta_p \leqslant H$ for all $p$, then $h v_1 \leqslant \sum_1^n a_p v_p \leqslant H v_1$ for all $n$. We have $$ \begin{aligned} a_1 & =s_1, a_2=s_2-s_1, \ldots, a_n=s_n-s_{n-1} \ \boldsymbol{B}n & =\sum_1^n a_p v_p=s_1 v_1+\sum{p=2}^n\left(s_p-s_{p-1}\right) v_p \ & =s_1\left(v_1-v_2\right)+\ldots+s_{n-1}\left(v_{n-1}-v_n\right)+s_n v_n \end{aligned} $$
数学代写|数学物理方法代写Mathematical Methods代考|PCS622 Extension of the M test
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
1-11. Functions of two variables. So far we have been considering sequences, which may be regarded as functions of one variable capable of taking only integral values, and functions of a continuous variable. In what follows we shall be concerned with what are essentially functions of two variables, which may be either integral or continuous. This introduces new complications when limiting processes are used, since it is not always obvious, or even true, that the same result will be obtained when the order of the limiting processes is changed. The simplest sufficient condition for the reversibility of limiting processes is provided by the following theorem on absolute convergence. 1-111. If $f(x, y)$ is a non-decreasing function of both $x$ and $y$ (either or both of which may take only integral values), and $$ \lim {x \rightarrow \infty} f(x, y)=g(y), \quad \lim {y \rightarrow \infty} f(x, y)=h(x), $$ then $$ \lim {y \rightarrow \infty} g(y)=\lim {x \rightarrow \infty} h(x), $$ in the sense that if either of the limits in (2) exists the other exists and the two are equal. Note first that $g(y)$ is a non-decreasing function of $y$. For if $y_2>y_1$ $$ g\left(y_2\right)-g\left(y_1\right)=\lim _{x \rightarrow \infty}\left{f\left(x, y_2\right)-f\left(x, y_1\right)\right} \geqslant 0 $$
数学代写|数学物理方法代写Mathematical Methods代考|Uniform convergence of sequences and series
1-112. Uniform convergence of sequences and series. The terms of a sequence $\left{f_n(x)\right}$ may be functions of a variable $x$. Then if the sequence converges for all values of $x$ in an interval, its limit is a function of $x$, say $f(x)$. If we choose an arbitrarily small positive $\epsilon$ we shall for any $x$ be able to choose $n(x)$ so that $\left|f_p(x)-f(x)\right|<\epsilon$ for all $p \geqslant n(x)$ because the sequence converges. In general the least value of $n(x)$ such that this is true will depend on $x$. But it may be possible to choose an $n$ independent of $x$ such that $\left|f_p(x)-f(x)\right|<\epsilon$ for all $p>n$ and for all $x$ in the interval. If this is possible for every $\epsilon, f_n(x)$ is said to be uniformly convergent to $f(x)$ in the interval. It can fail to be uniformly convergent if there is an $x$, say $c$, within or at the end of the interval such that if we take a succession of values of $x$, tending to $c$, the corresponding values of $n(x)$ for given $\epsilon$ tend to infinity. As $f_n(x)$ may be the sum of the first $n$ terms of a series, all these statements have immediate analogues for series $\Sigma u_n(x)$ over an interval of $x$. Thus the series $\Sigma x^n$ converges for all $x$ such that $0 \leqslant x<1$, but it is not uniformly convergent for all such $x$. For if we fix $\epsilon$ and then choose $n$ so as to make $$ x^n+x^{n+1}+\ldots+x^{n+p}=\frac{x^n\left(1-x^{p+1}\right)}{1-x} $$ less than $\epsilon$ for all $p \geqslant 1$ we must make $$ x^n<(1-x) \epsilon, $$ and therefore $$ n>\frac{\log {(1-x) \epsilon}}{\log x}, $$ which tends to infinity as $x$ tends to 1 . This series is therefore uniformly convergent in a range $a \leqslant x \leqslant b$, where $a$ and $b$ are fixed quantities between 0 and $+1$, since we can choose $n$ greater than the greater of the quantities $$ \frac{\log {(1-a) \epsilon}}{\log a}, \frac{\log {(1-b) \epsilon}}{\log b}, $$ and the same value of $n$ will then do for any intermediate value of $x$. It is convergent for any $x$ such that $-1<x<1$. But it is not uniformly convergent in the range $-1<x<1$ because, even though the signs < exclude the possibilities that $x$ may be actually $-1$ or $+1$, they permit any intermediate value, however close to 1 , and however we choose $n$ we shall always be able to find values of $x$ such that (3) is false.
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
数学代写|数学物理方法代写Mathematical Methods代考|MATH0056 Orders of magnitude
数学代写|数学物理方法代写Mathematical Methods代考|Orders of magnitude
1.08. Orders of magnitude. If as $x$ tends to a limit $\phi(x)$ tends to 0 or $\infty$, and $f(x) / \phi(x)$ is bounded, we say that $f(x)=O{\phi(x)}$, or that $f(x)$ is of the same order of magnitude as $\phi(x)$. If $f(x) / \phi(x) \rightarrow 0$ as $\phi(x) \rightarrow 0$ we write $f(x)=o{\phi(x)}$. If $f(x)$ is bounded we can write $f(x)=O(1)$. This notation must be distinguished from the common usage in physics, where we may say that the masses of Jupiter and Saturn are of the same order of magnitude, meaning roughly that they differ by not more than a factor 10 without there being any question of a limit. In the physical sense the quantities compared must have the same dimensions. This is not necessary in the mathematical sense. $x$ may, for instance, be a time-interval and $f(x)$ the distance travelled by a sound wave. Then $f(x)=O(x)$ because $f(x) / x$ is the velocity of sound and is supposed finite. Note that $$ O\left(x^m\right) O\left(x^n\right)=O\left(x^{m+n}\right), \quad o\left(x^m\right) O\left(x^n\right)=o\left(x^{m+n}\right) . $$
Functions of bounded variation. If the function $f(x)$ is defined in the closed interval $(a, b)$, and there is a number $M$ such that $$ v=\left|f\left(x_1\right)-f\left(x_0\right)\right|+\left|f\left(x_2\right)-f\left(x_1\right)\right|+\ldots+\left|f\left(x_n\right)-f\left(x_{n-1}\right)\right| \leqslant M, $$ for every subdivision $a=x_0<x_1<x_2<\ldots<x_{n-1}<x_n=b, f(x)$ is said to be of bounded variation in the interval $(a, b)$; and the upper bound of the sums $v$ for all possible selections of the subdivisions is called the total variation of $f(x)$ in the interval.* The total variation is of interest since it is related to the condition for existence of a Stieltjes integral (1-102), and to the existence of the length of a curve, and it is useful in the theory of Fourier series and Fourier integrals.
数学代写|数学物理方法代写Mathematical Methods代考|Leap at a discontinuity
1.094. Leap at a discontinuity. Let $f(x)$ be discontinuous at $a$ but bounded in an interval including $a$ as an interior point. Then for some positive $\delta, f(x)$ has upper and lower bounds $M, m$ in $(a-\delta, a+\delta)$. If $\delta^{\prime}<\delta$, the upper bound in $\left(a-\delta^{\prime}, a+\delta^{\prime}\right)$ cannot be greater, or the lower less, than in $(a-\delta, a+\delta)$. Hence the leap in $(a-\delta, a+\delta)$ has a non-negative limit as $\delta \rightarrow 0$. If this limit is zero the function is continuous at $a$; if positive, we call the limit the leap of the function at $a$.
If $f(x)=0(x<0), f(x)=1(x \geqslant 0)$, the leap at 0 is 1 . If $f(x)=0(x \neq 0), f(x)=1(x=0)$ the leap at 0 is again 1 . If $f(x)=\sin 1 / x$, the leap at $x=0$ is 2 , since values arbitrarily near 1 and $-1$ occur in any interval about 0. 1-10. Integration: Riemann, Stieltjes. Two different definitions of an integral will be used in this book.
Let $x_1, x_2, \ldots x_n$ be a set of increasing values of $x$ between $a$ and $b$, subject to all $x_{r+1}-x_r<\delta$ (we take when convenient $x_0=a, x_{n+1}=b$ ). Take in each interval a $\xi_r$, so that $x_r \leqslant \xi_r \leqslant x_{r+1}$; and form the sum $$ S_n=f\left(\xi_0\right)\left(x_1-a\right)+f\left(\xi_1\right)\left(x_2-x_1\right)+\ldots+f\left(\xi_n\right)\left(b-x_n\right) . $$ This sum will depend both on the values chosen for the $x_r$ and on those for $\xi_r$, unless $f(x)$ is constant; but if we take a sequence of values of $\delta$ tending to zero, taking at every stage $x_r$ and $\xi_r$ in accordance with the inequalities, and form the sum $S_n$ for each, these sums may tend to a limit, and this limit may be independent of the choice of the $x_r$ and $\xi_r$ at each stage. If so, this limit is called the Riemann integral and denoted by $$ \int_a^b f(x) d x $$
数学代写|数学物理方法代写Mathematical Methods代考|MATH0056 Orders of magnitude
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
Returning to the initial setting of a fibration the opposite extreme to the semiclassical case corresponds to a trivial fibration in the sense that $\phi$ has one fibre. Then the operators are simply smooth in the parameter $t$. This serves to emphasize that the map to $[0,1]$ is itself a fibration and this can be generalized – as in the setting of the Atiyah-Singer index theorem – to the case of a more general base and fibration with $X \times[0,1]$ replace by a fibration $\phi: \hat{X} \longrightarrow B$ with typical fibre $X$. This in turn can be generalized to the case of a $b$-fibration which allows degeneration, of a specific type, on the fibres. Rather than define this in general let me point to a specific type of example.
Suppose $M$ is a compact manifold with corners. It may have many boundary hypersurfaces but each has (by assumption) a defining function – a smooth nonnegative function on $M$ which vanishes precisely at the boundary hypersurface in question and has non-zero differential there. Then a total boundary defining function on $M$ is a product of such functions. More generally one can take a product of positive integral powers of such functions. The resulting function, which vanishes at every boundary point but is positive elsewhere, defines a b-fibration – a kind of collar decomposition – as a map to $[0, \epsilon]$ for $\epsilon>0$ small enough. The general case of a b-fibration is locally the product of such maps and an ordinary fibration. In any case there is a similar structure to the case of a fibration if vector fields tangent to all boundaries are considered and there is an algebra of pseudodifferential operators reducing to the fibre pseudodifferential operators for a fibration: $$ \mathcal{V}{\mathrm{b} / \phi}(M) \subset \rho^\alpha \operatorname{Diff}{\mathrm{b} / \phi}^(M ; W) \subset \rho^\alpha \Psi_{\mathrm{b} / \phi}^(M ; W) \subset \rho^\alpha \Psi_{\mathrm{b} / \phi}^{* *}(M ; W) $$ The last space here actually depends on a choice of the resolution of the fibre diagonal ([3]).
Example of this is provided by the blow-up calculus being developed with Pierre Albin and also the gluing calculus with Michael Singer which corresponds to gluing problems such as treated by Arezzo and Pacard [1]. Namely, from a manifold with an interior separating hypersurface a manifold with corners can be constructed with b-fibration which corresponds to the process of gluing a complete metric on one side of the hypersurface to an appropriate (often incomplete) metric on the other side. The b-fibration is of the type discussed above. This is also related to older work with Andrew Hassell and Rafe Mazzeo on the eta invariant.
The two types of calculus above, corresponding to a b-fibration, where the vector fields degenerate only at a submanifold in the boundary, and the adiabatic case where the degenerate to be tangent to the fibres of a fibration can be combined. Rather than set this up in general – corresponding to iterated b-fibrations where there are finer fibrations over (some of) the boundary hypersurfaces of the first b-fibration – let me simply indicate an example which arises from a question of Atiyah.
Every compact manifold $M$ carries a Morse function $f: M \longrightarrow[a, b]$. This can be thought of as a generalization of a fibration over the circle – the setting considered by Witten in [4]. There are singular fibres but they are isolated and of ‘minimal singularity’. In particular the singular points, where the differential of $f$ vanishes, are themselves isolated. To construct an adiabatic limit of this b-type, first replace $M$ by the manifold with boundary in which the critical points of $f$ are blown up radially, $M_{\mathrm{C}}=[M ; \mathrm{Cp}]$ to which $f$ lifts as a smooth function. The singular fibres of $f$ are resolved in the sense that they are each the union of a boundary hypersurface and an embedded (generally non-connected) submanifold SF which meets this boundary transversally. The full space with b-fibration we consider is $$ \left[M_{\mathrm{C} P} \times[0,1]_t ; \mathrm{SF}\right] \longrightarrow[0,1]_t . $$ The additional condition imposes on vector fields (and hence differential operators) corresponding to the adiabatic limit is that over $t=0$ they should be tangent to the boundaries, to the regular fibres of $f$ and to the fibres of the blow-down map for the blow-up of SF.
Atiyah’s question is whether for a Dirac operator on the total space one can find a formula for the index (which of course is known) in terms of the spectral flow of the induced Dirac operators on the fibres, between the singular values (likely regularized in some way) with perhaps some ‘jump terms’ across the singular fibres. For the moment I only know how do do this after perturbing the operator by a smoothing operator associated to the calculus that I have implicitly described above. To give a more realistic answer requires a better understanding of the behaviour of the eta invariant.
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
数学代写|数学物理代写Mathematical Physics代考|PHZ3113 Degenerate Fibres: Analytic Case
数学代写|数学物理代写Mathematical Physics代考|Degenerate Fibres: Analytic Case
One natural question is what happens to the adiabatic limit if the form of the metric is generalized. This is related to issues below and there are several levels of ‘relaxation’ of the conditions. One might first consider perturbations of first order in $t$ which include cross terms between base and fibre such as $t d y \otimes d z$. Note that it does not really make sense to want these to have coefficients which only depend on the base variables. The effect of adding such a term can be quite dramatic, since it can change the invertibility properties of the adiabatic model operators although this is still an elliptic suspended family – the null spaces may no longer form a bundle, as they may not be smooth over the base. If the failure to be smooth is itself reasonably smooth, as discussed below, then something can be done. I don’t really know what happens to the null space in the general case – if anyone wants to try to work it out they are welcome to try and I am interested to discuss it!
Another possible ‘perturbation’ is to replace $h$ by a basic tensor – that is, to let its coefficients vary on the fibre. This is actually less of a problem than the introduction of cross terms and leads to a very similar structure, but the details have not been written up as far as I know.
Next consider the effect of adding a potential, for simplicity in the case of the Laplacian on functions. The simplest case is when the potential is real and nonnegative. If it is non-zero on any fibre then the model on that fibre is invertible. The opposite extreme to uniform degeneracy is when $V$ vanishes on isolated fibres and has non-zero Hessian (in the base variables) at every point on those fibres
Then $\Delta_t+V$ has an inverse with a weaker (optimal) bound than in a case such as (4) when the model operators are invertible $$ \left|\left(\Delta_t+V\right)^{-1}\right|_{L^2} \leq C t $$ In this case there are additional model operators at the singular fibres which are harmonic oscillators.
数学代写|数学物理代写Mathematical Physics代考|Eigenvalues for Triangles
One geometric setting closely related to the perturbation by a potential with isolated minima is the vertical collapse of a manifold with boundary. For example, if one takes a region in the plane between two $2 \pi$-periodic smooth curves, $\Omega=\left{(x, y) \in \mathbb{R}^2 ; L(x)<y<U(x)\right}$, and ‘collapses’ it by vertical scaling to $\Omega_t=\left{(x, y) \in \mathbb{R}^2 ; L(x)<y / t<U(x)\right}, 0<t \leq 1$, then consider the behaviour of the eigenvalues for the Dirichlet, or Neumann, problem. For the Dirichlet problem the smallest eigenvalue has an asymptotic expansion related to that above, in particular there are harmonic oscillator models, provided $U(x)-L(x)$ only has non-degenerated maxima.
The general problem of the behaviour of the eigenvalues for the Dirichlet problem for triangles as functions on moduli space remains open and certainly there is behaviour similar to this under vertical collapse, except that the harmonic oscillator is replaced by its ‘one-sided’ cousion, Airy’s operator.
数学代写|数学物理代写Mathematical Physics代考|PHZ3113 Degenerate Fibres: Analytic Case
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
数学代写|数学物理代写Mathematical Physics代考|PHYS3260 Adiabatic Limit with Isolated Degenerate Fibres
数学代写|数学物理代写Mathematical Physics代考|Adiabatic Limit with Isolated Degenerate Fibres
Recall that the basic set up of an adiabatic limit corresponds to a fibration of manifolds – a submersion between compact connected manifolds $\phi: X \longrightarrow Y$ with typical fibre $Z$. On the total space $X$ one can consider a family of ‘adiabatic metrics’ $g_t=\phi^* h+t^2 \mu$ where $h$ is a metric on the base (it could also depend on $t$ ) and $\mu$ is a smooth symmetric 2 -cotensor on the fibres which restricts to be positive definite, and hence a metric, on the fibres. In fact I think it is more natural to consider a family of metrics such as $t^{-2} g_t$ where the fibres are of more-or-less constant size and fixed vectors in the base get ‘large’ which means the base is ‘slow’ (hence the term adiabatic). I believe Witten [4] was the first to consider global analysis related to such metrics when he examined the behaviour of the eta invariant for the particular case of a fibration over a circle.
Note that this setting is more general than a Riemannian submersion, which corresponds to the case that $\mu$ has rank exactly equal to the dimension of the fibres at each point and I will mention some other possible generalizations below.
For $t>0$ nothing much is happening, one just has a smooth family of metrics and $t$ is simply a parameter. On the other hand one can view the singular limit at $t=0$ as imposing a ‘geometry’. Then $t$ is no longer a true parameter but should be included in the analysis, so instead of $X$ consider the space $X \times[0,1]$ where the iterval corresponds to $t$. The basic object to consider is the space of smooth vector fields on $X \times[0,1]$ $$ \mathcal{V}_{\mathrm{a}}(X)=\left{V ; V t=0, V \text { tangent to } \phi^{-1}(y) \text { at } t=0\right} \text {. } $$
Although the ‘main’ solvability issue above appears to be the invertibility of the fibre Laplacian. Really, it is not quite this which is involved, or rather it is a little more than that. Namely what we actually get is a suspended Laplacian. This is the ‘adiabatic symbol’. Note that the ‘semiclassical’ case is when $\phi$ is the identity. Then the adiabatic, or semiclassical, symbol is the ‘full symbol’ of the Laplacian. In case of a general fibration $\phi$ it is a bundle, over $T^* Y$, of operators. For each point in the base we get a differential operator (on some form bundle) over the space $T_y Y \times Z_y$, where $Z_y$ is the fibre above $y$. This is a second order elliptic differential operator and is translation-invariant in $T_y Y$. We can take the Fourier transform and hence get a differential operator on $Z_y$ which is polynomial in $T_y^* Y$. The main issue is then is the invertibility of this ‘suspended’ family (suspended in the topological sense of having some Euclidean variables added). For an adiabatic metric such as described above this turns out to be straightforward, the null space can be decomposed (over forms from the base) in terms of the fibre null spaces which form an ordinary vector bundle, the forms on $Y$ twisted by the flat bundles of fibre harmonic forms.
So for the Laplacian on forms this model operator can never be fully invertible the constants in degree 0 always intervene. For other similar problems it can. For instance if the fibres are manifolds with boundary and one considers the Dirichlet boundary condition, then the adiabatic model operators are fully invertible and in consequence the full Laplacian is also invertible uniformly:- $$ \left|\Delta_{t, \text { Dir }}^{-1}\right|_{L^2} \leq C t^2 \text { as } t \downarrow 0 . $$ Then a more interesting question, touched on below, is the behaviour of the lowest eigenvalue and eigenstate.
数学代写|数学物理代写Mathematical Physics代考|PHYS3260 Adiabatic Limit with Isolated Degenerate Fibres
对于$t>0$没有发生什么事情，我们只是有一个平滑的度量家族，$t$只是一个参数。另一方面，人们可以把$t=0$上的奇异极限看作是施加了一个“几何”。那么$t$就不再是一个真参数，而是应该被包含在分析中，因此考虑空间$X \times[0,1]$而不是$X$，其中迭代值对应于$t$。要考虑的基本对象是$X \times[0,1]$ $$ \mathcal{V}_{\mathrm{a}}(X)=\left{V ; V t=0, V \text { tangent to } \phi^{-1}(y) \text { at } t=0\right} \text {. } $$
上光滑向量场的空间
数学代写|数学物理代写数学物理代考|均匀简并
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尽管上述“主要”可解性问题似乎是纤维拉普拉斯式的可逆性。实际上，问题并不完全在于此，或者更确切地说，问题远不止于此。也就是说，我们得到的是一个悬挂的拉普拉斯式。这是绝热符号。请注意，“半经典”情况是在$\phi$是标识时。那么绝热符号，或者说半经典符号，就是拉普拉斯方程的”完整符号”对于一般的纤维化，$\phi$是操作符的一个包，超过$T^* Y$。对于基数中的每个点，我们在空间$T_y Y \times Z_y$上得到一个微分运算符(在某种形式束上)，其中$Z_y$是$y$上面的光纤。这是一个二阶椭圆微分算子，在$T_y Y$中是平移不变的。我们可以进行傅里叶变换得到$Z_y$上的微分算子它是$T_y^* Y$上的多项式。主要的问题是这个“悬浮”族的可逆性(悬浮在添加了一些欧几里德变量的拓扑意义上)。对于如上所述的绝热度规，这是很简单的，零空间可以根据纤维零空间分解(从基的形式)，纤维零空间形成一个普通向量束，$Y$上的形式被纤维谐波形式的平面束扭曲
所以对于拉普拉斯on形式这个模型算子永远不可能完全可逆0次的常数总是会介入。对于其他类似的问题，它可以。例如，如果纤维是有边界的多形，并且考虑Dirichlet边界条件，那么绝热模型算子是完全可逆的，因此完全拉普拉斯算子也是一致可逆的:- $$ \left|\Delta_{t, \text { Dir }}^{-1}\right|_{L^2} \leq C t^2 \text { as } t \downarrow 0 . $$ 那么下面要讨论的一个更有趣的问题是最低特征值和特征态的行为
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
1.0621. The Heine-Borel theorem. If every point of a closed interval $(a, b)$ is within some interval I of a family $F$, then there is a finite subfamily of $F$ such that every point of $(a, b)$ is within at least one interval of the subfamily. We say that $I$ covers $(c, d)$ if every point of $(c, d)$ is an interior point of $I$ (i.e. not an end-point).
There may be an interval $I$ belonging to $F$ that covers the whole of $(a, b)$. If so there is nothing to prove. If not, bisect $(a, b)$. There may be a pair of intervals $I_{1}, I_{2}$ such that every point of $\left(a, \frac{1}{2} a+\frac{1}{2} b\right)$ is interior to $I_{1}$ and every point of $\left(\frac{1}{2} a+\frac{1}{2} b, b\right)$ to $I_{2}$. If either half is not included in an interval $I$, bisect that half. We say that in a finite number of steps we shall arrive at a stage where every portion of $(a, b)$ lies within at least one interval $I$. For if not, the successive bisection of intervals will give a sequence of intervals, each part of the preceding one, and each half the length of the preceding one, and none of them included in an $I$. Such a sequence forms a nest of intervals and identifies a number $x_{0}$ common to all its members. But by hypothesis $x_{0}$ is interior to an $I$, say $I_{0}$, and hence there is a positive $\delta$ such that all points of $\left(x_{0}-\delta, x_{0}+\delta\right)$ are in $I_{0}$. Therefore all intervals of the nest whose lengths are less than $\delta$ are included in $I_{0}$, and we have a contradiction. Hence every division of $(a, b)$ is wholly interior to some $I$. Taking for each division an $I$ that includes it we have the theorem.
A slight modification is often made where an end-point, say $a$, is an end-point of an interval of the family, say $I_{a}$, closed at $a ; I_{a}$ is still supposed of non-zero length $\delta_{a}$. Then $a$ is interior to the interval $J_{a}\left(a-\frac{1}{2} \delta_{a}, a+\frac{1}{2} \delta_{a}\right)$, and the argument applies to the set of intervals $J$, where $J$ is the same as $I$ except that $I_{a}$ is replaced by $J_{a}$; every point of $(a, b)$ is an interior point of at least one $J$. But then the theorem follows with the modification that $a$ may be an end-point of $I_{a}$ or $b$ of $I_{b}$ provided that $I_{a}$ has $a$ as a member and $I_{b}$ has $b$.
The theorem gives the Bolzano-Weierstrass theorem (1-034) as a special case. If possible, let $(a, b)$ contain no limit-point of the set. Then every point of $(a, b)$ is in an interval $I$ containing not more than one member of the set. Hence $(a, b)$ can be covered by a finite set of such intervals $I$ and therefore contains only a finite number of members of the set of points considered, contrary to hypothesis.
In the argument as we have stated it the only intervals bisected at each stage are those not already covered by an $I$. We could, however, equally well bisect all the intervals. For if $I$ covers $(c, d)$ it covers both halves of it. Hence $(a, b)$ in the conditions stated can be divided into a finite set of equal intervals each covered by an $I$.
1.0622. The modified Heine-Borel theorem. In the Heine-Borel theorem the intervals $I$ may be specified by any rule so long as each is of non-zero length and every point of $(a, b)$ is an interior point of at least one of them (except that $a$ and $b$ may be end-points). Sometimes, however, a further restriction is made, according to which each point $x$ of $(a, b)$ specifies an $I_{x}$, of which $x$ is an interior point. Then the following theorem holds. Suppose that every point $x$ of $a \leqslant x \leqslant b$ is within an interval $I_{x}\left(x-\delta_{x}, x+\eta_{x}\right)$, where $\delta_{x}>0$, $\eta_{x}>0$, except that $I_{a}$ may be $a \leqslant x<a+\eta_{a}$ and $I_{b}$ may be $b-\delta_{b}<x \leqslant b$; then $(a, b)$ may be divided into a finite set of intervals such that each interval is part of the $I_{x}$ defined for some point of that interval. The proof is by successive bisection as before. Assuming the theorem false, we establish the existence of a nest of intervals converging to some $x_{0}$, such that none is part of $I_{x}$ for any $x$ within that interval; but all of them less than a certain length are parts of $I_{x_{0}}$, and contain $x_{0}$, and we have a contradiction. In this case, however, it does not follow that $(a, b)$ can be divided into equal intervals with the required property. If $(c, d)$ is covered by $I_{x}$, where $x$ is in $(c, d), x$ can be interior to only one half of $(c, d)$; then the other half is not necessarily covered by an $I_{v}$, when $y$ is now restricted to be in that half.
An important application is to differentiable functions. Let $f(x)$ be differentiable at all points of $a \leqslant x \leqslant b$; this says that for any $\omega$, for any $x$ in $(a, b)$, there is a positive $\delta(\omega, x)$ such that for $|h|<\delta \quad\left|f(x+h)-f(x)-h f^{\prime}(x)\right|<\omega|h|$ (1) Then $(a, b)$ can be divided into a finite set of intervals $\left(x_{r}, x_{r+1}\right)$ such that for all $x$ of $\left(x_{r}, x_{r+1}\right)$ $$ \left|f(x)-f\left(\xi_{r}\right)-\left(x-\xi_{r}\right) f^{\prime}\left(\xi_{r}\right)\right|<\omega\left|x-\xi_{r}\right| \text {, } $$ $\xi_{r}$ itself being a point of $\left(x_{r}, x_{r+1}\right)$. For any fixed $\eta$ (1) remains true if all $\delta(\omega, x)$ are restricted to be $\leqslant \eta$. Then all $x_{r+1}-x_{r}$ will be $\leqslant 2 \eta$.
Heine* proved that a continuous function is uniformly continuous (1.071) by what was essentially a method of Dedekind section, capable of being used to prove the general Heine-Borel theorem and so used in Lebesgue’s proof. The specific use of overlapping intervals is due to Borel $\dagger$, the form of the Heine-Borel theorem given here to W.H.Young.f The bisection method was used by Bolzano; Goursat (see 11.043) used it in an important simplification of the conditions for Cauchy’s theorem, in which he recognized the effect of the restriction when each section is required to contain a point $x$ with which the $I_{x}$ covering that section is associated. He did not, however, give the general form of the modified theorem or comment on the possibility of proving the main theorem by the same method. This was first done by H. F. Baker in a note reported in title only. $\S$
数学代写|数学物理方法代写Mathematical Methods代考|MATH0056 The Heine-Borel theorem
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
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