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For a measurable set $A\subseteq {\mathbb{R}}^n$, its density at $x$ with respect to a locally finite measure $\mu$ iswhen this limit exists.

A function $f:(X,\mathcal{A})\to (Y,\mathcal{B})$ between measurable spaces is measurable when $f^{-1}(B)\in \mathcal{A}$ for every $B\in \mathcal{B}$.

Projection onto the mean-variance portfolio $V^{-1}b$ leaves residual covariance $V-b{b}^T/(b^T{V}^{-1}b)$.

A portfolio dominates another when it has no smaller mean and no larger variance, with at least one strict improvement.

Let $\xi \sim N(b,\Sigma )$, where $b\ne 0$ and $\Sigma$ is positive definite. For any increasing concave objective of $m+{\theta }^T\xi$, every unique optimal portfolio has the formIndeed, the component orthogonal to ${\Sigma }^{-1}b$ in the $\Sigma$ inner product contributes independent mean-zero Gaussian risk without changing the mean. Removing it cannot reduce expected concave utility. A negative coefficient is dominated by the corresponding positive coefficient, which has the same variance and a larger mean.

Without a risk-free intercept or constraints, every undominated linear portfolio lies on the nonnegative ray through $V^{-1}b$.

A linear anisotropic medium has constitutive relations $D_i={\epsilon }_{ij}{E}_j$ and $B_i={\mu }_{ij}{H}_j$. Symmetric, time-independent tensors make the field-energy density a quadratic form.

Here permittivity is a tensor, so the electric displacement need not be parallel to the electric field.

If a function is continuous on the closure of a bounded plane domain and holomorphic inside, the compact closure supplies a point of maximum modulus. Unless the function is constant, the maximum modulus principle places every maximum on the boundary.

In a regular $p$-parameter model with one-observation Fisher information $I({\theta }_0)$,

For independent exponential observations with rate $\theta$, the MLE is $1/\overline{X}$ and has asymptotic variance ${\theta }^2/n$.

For $X\sim \mathrm{Bin}(n,\theta )$, maximizing ${\theta }^X(1-\theta {)}^{n-X}$ gives

The log-likelihood is the natural logarithm of the likelihood function:For positive finite likelihood, the natural logarithm is a strictly increasing function, so the maximizing parameter values are unchanged. Set $\ell =-\infty$ where $L=0$; the same ordering then includes zero-likelihood boundaries whenever a positive finite maximum exists. For independent and identically distributed random variables, a product becomes a sum,This makes derivatives simpler and avoids forming products of tiny numbers in floating-point arithmetic. These equal-maximizer and additive formulas underlie the worked calculations in MIT 18.05, Reading 10b, section 3.1.

For the ordered sample of eight successes and two failures in likelihood function, on $0<p<1$,The score equation ${\ell }^{\prime }(p)=0$ gives $8(1-p)=2p$, hence $\hat{p}=8/10=0.8$. The negative second derivative proves strict concavity, so this is the unique interior maximum. Both counts are positive, so $L(0)=L(1)=0$, while $L(p)>0$ inside: the same point is the unique global maximum on $[0,1]$. Counting only successes adds $\log 45$ to the log-likelihood, leaving both derivatives and the MLE unchanged. For all-success or all-failure samples, the boundary analysis in maximum-likelihood estimator applies instead of an interior score equation.

The two panels below compare $k=2,5,8$ successes out of ten Bernoulli trials. Their peaks are at $\hat{p}=k/10$. The left panel uses relative height; the right panel uses the log-likelihood difference from each peak. These are the same comparison on two scales:A difference of $-2$ means relative height $e^{-2}$, and a difference of $-10$ means $e^{-10}$; the log-likelihood scale makes small relative heights easier to see. Neither vertical axis is a parameter probability density function. The zero likelihood endpoints have log-likelihood $-\infty$ and are outside the finite vertical range in the right panel.

Multiplying a likelihood function by a positive parameter-independent constant adds a constant to its log-likelihood. Subtracting its peak removes that constant. This is why the ordered-data and count-data curves coincide in this figure after rescaling. When differentiable, the gradient of the log-likelihood is the score function; the negative Hessian matrix is the observed information. A solution of a score equation still needs checking against other stationary points and boundaries: maximizing the log-likelihood does not automatically make it concave.

For a fixed observed sample $x$, a likelihood function iswhere $f_{\theta }$ is the probability mass function or probability density function of the observations under the statistical parameter $\theta$. All densities must be defined relative to the same parameter-independent dominating measure. The sample is fixed; the parameter is the function's argument. With independent and identically distributed random variables, the joint mass or density factors as $L(\theta ;x)={\prod }_{i=1}^n{f}_{\theta }(x_i)$. This definition and the distinction between sampling probability and parameter uncertainty are explained in MIT 18.05, Reading 10b, sections 3 and 3.2.

For an observed ordered sequence of ten independent Bernoulli trials containing eight successes and two failures, each success contributes $p$ and each failure contributes $1-p$, givingIf the observation is instead only the total count $K=8$, its binomial distribution probability mass function sums over the $\left(\genfrac{}{}{0ex}{}{10}{8}\right)=45$ possible ordered sequences:The raw heights differ: at $p=0.8$ they are approximately $0.0067108864$ and $0.301989888$. Their maximum-likelihood estimates are identical. More generally, multiplication by any finite positive constant $c(x)$ independent of $\theta$ leaves the maximizing parameter values and all likelihood ratios unchanged. Multiplication by a parameter-dependent factor can change the MLE; dropping such a factor is not justified. Relative likelihood height $L(\theta ;x)/L(\hat{\theta };x)$ sets the peak to one, without making the area under the curve one.

Normalization of $f_{\theta }$ is over possible data values for each fixed parameter. It imposes no requirement that ${\int }_{\Theta }L(\theta ;x)d\theta =1$. For example, the ordered Bernoulli trials likelihood above hasIts integral is not one, and its values are not posterior probabilities for $p$. When the Bayesian posterior is defined, Bayes' theorem multiplies the likelihood by a prior density and normalizes that product. Choosing to normalize a likelihood function alone is an additional choice, such as using a uniform prior in those parameter coordinates, rather than part of the definition of likelihood.

For observations with a probability density function, the probability of an exact sample point is zero. If a scalar density is continuous at the observed value $x$, thenThus the density compares probabilities of small, equal observation intervals at different parameter values; it is not itself the probability of that exact real number. The equal-width factor is independent of the parameter. This gives visual intuition for density likelihoods, as in MIT 18.05, Reading 10b, section 4. The normal distribution example in maximum-likelihood estimator makes the comparison explicit. If only rounded intervals are recorded, their integrated probabilities give the exact observation likelihood; using midpoint densities is an approximation.

An ideal $\mathfrak{m}$ of a commutative unital ring $R$ is maximal exactly when $R/\mathfrak{m}$ is a field.

The maximal spectrum $\mathrm{MaxSpec}R$ is the set of maximal ideals of a commutative ring $R$.

The residue field at a maximal ideal $\mathfrak{m}$ is the quotient field $\kappa (\mathfrak{m})=R/\mathfrak{m}$. At a prime ideal $\mathfrak{p}$, the residue field is the fraction field of $R/\mathfrak{p}$.

After neglecting the decaying mode, the linear growth between conformal times ${\tau }_1$ and ${\tau }_2$ is

In cosmic time during matter domination, $a(t)\propto t^{2/3}$ and the pressureless density contrast obeysIts growing and decaying solutions are $t^{2/3}$ and $t^{-1}$.

Forthe derivative rule givesand hence $Y(s)=(sI-A{)}^{-1}(y_0+G(s))$ wherever the transforms converge and $sI-A$ is invertible.

If $A$ is an $m$-by-$n$ matrix and $B$ is an $n$-by-$m$ matrix over the same field, both products are square matrices andThe products can even have different sizes: for $A=(12)$ and $B=(34{)}^T$, $AB=(11)$ and $BA=\left(\begin{array}{cc}3& 6\\ 4& 8\end{array}\right)$, but both have trace $11$.^[ref]

Grouping all but one factor proves that $\mathrm{tr}(A_1{A}_2\cdots A_k)$ is unchanged by a cyclic permutation whenever the cyclic products are defined and square. In particular, every commutator has trace zero, since $\mathrm{tr}(AB-BA)=0$ by linearity.

Cyclic rotation does not allow arbitrary reordering. Forone has $ABC=A$, so $\mathrm{tr}(ABC)=1$, but $AC=0$, so $\mathrm{tr}(ACB)=0$. Rotating $ABC$ to $BCA$ or $CAB$ preserves its trace; swapping just $B$ and $C$ need not.