At zero the indicial equation is $\sigma (2\sigma -1)=0$. The two Frobenius series have exponents zero and one half, with recurrenceThe denominators never vanish for these exponents. Unless a series terminates its coefficient ratio tends to $-1/2$, giving convergence within distance two of zero. If $\alpha =m$ is a positive integer, the exponent-zero series terminates at degree $m$. The substitution $t=z+1$ gives the standard Chebyshev differential equation, and this solution is a multiple of $T_m(z+1)$. For half-integer $\alpha$, termination can instead occur in the power-series factor of the exponent-one-half solution.

A field has characteristic three when $1+1+1=0$. The binomial identity then becomes $(a+b{)}^3=a^3+b^3$, and $X^3-1=(X-1{)}^3$. Thus a cyclic shift operator on three coordinates has a repeated-root minimal polynomial in this characteristic and cannot be diagonalizable.

In the conjugacy class order $1,(123),(12)(34),(12345),(13452)$, the five complex irreducible characters of the alternating group $A_5$ areThe class sizes are $1,20,15,12,12$. Restricting the degree-four and degree-five irreducible characters of $S_5$ gives rows of squared character inner product one. The degree-six restriction has squared norm two and is orthogonal to those rows and the trivial row, so it splits into the two remaining irreducibles. The sum of squared degrees forces their degrees to be three. Orthogonality forces their last two values to have sum one and product minus one, producing the displayed roots. Since a normal subgroup is a union of classes and its order divides sixty, these class sizes also prove the Simplicity of the alternating group A5.

A group of order thirty-six with a normal $C_3^2$ and a cyclic quotient of order four acting freely on its eight nonzero elements has six conjugacy classes of sizes $1,4,4,9,9,9$. Its four linear characters are lifted from the cyclic quotient. The remaining irreducible degrees are four and four, by the sum of squared degrees. Column orthogonality forces the last three values of both degree-four rows to vanish. The square of the order-four action is inversion on $C_3^2$, making the first two nontrivial class values real. Orthogonality then gives sum minus one and product minus two, so the rows are $(4,1,-2,0,0,0)$ and $(4,-2,1,0,0,0)$. This illustrates why reality must be justified when reconstructing a character table from orthogonality.

For the nontrivial action of $C_3$ on $C_7$, the conjugacy class sizes are $1,7,7,3,3$. Three linear characters factor through the quotient $C_3$, and the remaining two irreducible characters have degree three. Put $\omega =e^{2\pi i/3}$ and let $\alpha ,\beta =(-1\pm i\sqrt{7})/2$. With suitable class labels the table isOne can derive the last two rows by inducing nontrivial characters of $C_7$ along the two orbits of the order-three automorphism on their exponents. Their degree-three orbit sums vanish off $C_7$. On a generator they give the two sums of seventh roots over quadratic residues and nonresidues. Their sum is $-1$ and their product is $2$: the nine pairwise exponent sums between the two triples contain zero three times and every nonzero residue once. Thus they are the roots of $X^2+X+2$. Alternatively column orthogonality forces the zeros on the centralizers of order three, and orthogonality with the identity column interchanges $\alpha$ and $\beta$ in the last row. This finite table illustrates how centralizer sizes and a small number of given values determine missing character values.

For a finite group, the sum of products of irreducible character values is zero when $g,h$ are not conjugate, and equals $|C_G(g)|$ when $g,h$ are conjugate. To prove this from row orthogonality relations for a character table, choose representatives $g_j$ of its conjugacy classes and form $U_{ij}=\sqrt{|g_j^G|/|G|}{\chi }_i(g_j)$. Row orthogonality says $U{U}^{\ast }=I$. There are equally many irreducible characters and conjugacy classes, so this is a square unitary matrix and $U^{\ast }U=I$ as well. Its column equations give the result, using $|G|/|g_j^G|=|C_G(g_j)|$.

Combine finitely many discrete memoryless channels with disjoint input alphabets and disjoint output alphabets into a block-diagonal channel. The block label $B$ is determined by both input and output. If it has law $\alpha$, the mutual information is at most $H(\alpha )+{\sum }_i{\alpha }_i{C}_i$, attained by using each block's capacity-achieving conditional input law. Put $S={\sum }_i{2}^{C_i}$ and $r_i=2^{C_i}/S$. The expression equals ${\log }_{2}S-d(\alpha ,r)$, so Gibbs inequality proves the displayed capacity, attained for block probabilities ${\alpha }_i=r_i$. Even blocks of capacity zero contribute distinguishable block choices; three disjoint zero-capacity blocks give capacity ${\log }_{2}3$.

Suppose one row of a finite discrete memoryless channel is a convex combination of other rows: $W(\cdot \mid x_0)={\sum }_j{\alpha }_{j}W(\cdot \mid x_j)$. Deleting input $x_0$ leaves the channel capacity unchanged. Given any input law with mass $t$ on $x_0$, move that mass to the other symbols in proportions ${\alpha }_j$. The output law stays the same. By the concavity of information entropy, $H(W(\cdot \mid x_0))\ge {\sum }_j{\alpha }_{j}H(W(\cdot \mid x_j))$, so this replacement cannot increase conditional entropy $H(Y\mid X)$ and cannot decrease mutual information. Taking maxima proves the claim; the smaller channel is also trivially a restriction of the original one. If the entropy inequality is strict, every positive use of the redundant row is strictly suboptimal compared with its redistribution.

Pass the output of a discrete memoryless channel of channel capacity $C$ through an independent finite-alphabet erasure channel with erasure probability $p$. Its capacity becomes $(1-p)C$.

To prove it, let $X$ be the original input, $Y$ the original output, $E$ the erasure indicator, and $Z$ the final output. Since the erasure marker is distinguishable, $E$ is determined by $Z$. It is independent of $X,Y$, so the chain rule for mutual information gives $I(X;Z)=I(X;E,Z)=I(X;Z\mid E)=(1-p)I(X;Y)$. Maximization over the same input distributions proves the formula. For $p<1$ the maximizing distributions are exactly the original channel's maximizers. For $p=1$, every input distribution attains the zero capacity. Input-dependent erasure need not satisfy this scaling identity.

Adjustment of cerebral vascular resistance that moderates changes in blood flow as perfusion pressure varies. It has finite physiological limits and is not a guarantee of constant flow.

An arterial connection at the base of the brain linking anterior and posterior circulations; anatomical variants affect its collateral capacity.

Cerebral perfusion pressure is the pressure difference supporting cerebral blood flow. A common approximation compares arterial pressure with intracranial pressure, while the effective downstream constraint can depend on venous conditions. The pressure difference alone does not determine flow without vascular resistance.

The blood-brain barrier is a specialized vascular interface restricting and regulating exchange between blood and neural tissue. Endothelial junctions, transport systems and interactions with supporting cells contribute. It is selective rather than an absolute seal against every substance.

The superior cerebellar peduncle is a major cerebellar connection carrying much output from deep cerebellar nuclei toward more rostral targets, along with selected input pathways. Its relation to the superior medullary velum helps orient the roof of the fourth ventricle.

The inferior cerebellar peduncle connects the cerebellum with medullary and spinal-associated pathways. It carries several sensory and olivary inputs together with some outputs. Its mixed composition differs from a single dedicated sensory nerve.

The arithmetic mean of a finite nonempty collection of vertices, treating each vertex as an equal point mass. For a polygon, this need not equal the uniform-area centroid of its interior. A translation or linear map commutes with this average; consequently rotating all vertices about their vertex centroid leaves that centroid fixed.

For the canonical hyperbolic partial differential equation $V_{\xi \eta }=0$, every solution is $V=g(\xi )+h(\eta )$. Data on the characteristic $\eta ={\eta }_0$ determine $g$ up to an additive constant, but only determine $h$ at one point. If a transverse derivative is also prescribed, it must satisfy $V_{\eta }(\xi ,{\eta }_0)=h^{\prime }({\eta }_0)$, independent of $\xi$. A varying prescribed value is incompatible; a constant compatible value still leaves the rest of $h$ arbitrary. Thus characteristic Cauchy data can produce nonexistence or nonuniqueness. A derivative in original coordinates must first be transformed by the chain rule before applying this compatibility test.

Hyaline cartilage is cartilage with a relatively uniform-looking matrix at the light-microscope scale. It supports structures and provides low-friction articular surfaces at many joints. The visual name does not mean the matrix lacks molecular or fibrillar organization.