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Assume condition (1). For fixed , the supremumis nondecreasing as increases. Therefore its limit as exists in the extended real numbers and equals . Because , it is at least . Condition (1) says precisely that this limit is .
It follows that for every there is with . Fix and with . Taking produces an open ball on which . Thus is an open set, and its complementThis proves (1) implies (2). The property being characterized is upper semicontinuity.
The two boundary circles meet at : subtracting their equations gives , and either equation then gives . Send these points to zero and infinity by the Möbius transformationThe point is on the boundary, so is holomorphic throughout . Write . Substituting the inverse gives the exact equivalencesThusThere is no missing point from the inverse here: lies outside this sector. The image is therefore exactly the stated sector, rather than merely a subset of it.
Rotate by to obtain the first quadrant, and take the power using its holomorphic logarithm with . The required map isHere the principal logarithm works because the expression inside it lies in the first quadrant. It multiplies its argument by , giving , and maps its modulus to , which ranges over all positive real numbers.
For explicit surjectivity and injectivity, given with , definewhere uses . Then , so and , and these are inverse operations. Each stage is holomorphic with nonzero derivative, proving that is the requested conformal bijection.
For nonzero meridional wavenumber,At least one growing zonally varying mode can exist only ifThe upper value is a supremum approached as , where the growth itself tends to zero. Thus not every permitted meridional wavenumber has the same fastest-growing zonal scale; nonzero restricts the available band of and weakens the growth factor at fixed .
Using , not the tabulated squared value itself, givesThe planetary radii put these into angular context: is about 280 for Earth and 149 for Jupiter, corresponding to minimum meridional angular wavelengths about and , respectively. They are reciprocal-length wave numbers; the dimensionless angular counts should not be substituted for in the Cartesian dispersion relation. If a channel admits a half-wave across its width, the corresponding minimum width is rather than a full wavelength.
The oceanic scale is mesoscale, while the much larger deformation length makes the Jovian modes thousands of kilometres across. Both are small fractions of the planetary radius, consistent with using a local patch as a first approximation. Real boundaries, planetary-vorticity gradients, varying stratification and jets can change these limits; this simple model alone does not predict the complete observed flow pattern.
Finally, the source's phrase must be interpreted in the mean-flow frame. The actual criterion is , or . In the original frame has nonzero real part, so is generally complex, not an ordered negative real number. For example, at and , and . This is an unstable wave even though the literal printed inequality is inapplicable.
The two conditions for a basis of a topology both hold. First, every real number belongs to , so the proposed basic intervals cover the real line. Second, ifthen , andis itself one of the proposed basic intervals. Thus every point in an intersection of two basic sets has a basic neighbourhood contained in that intersection.
Consequently the arbitrary unions of members of , including the empty union, form a topology . The basis of a topology conditions give closure under finite intersections; arbitrary unions are built into the definition. This is the lower limit topology, also called the Sorgenfrey line.
In binary long division, first double the divisor until it exceeds
x, then halve it successively. Keep p = 2^j and d = y*p, so no exponentiation is needed inside a loop:q := 0; r := x;
j := 0; p := 1; d := y;
WHILE d <= x DO
d := 2*d;
p := 2*p;
j := j+1
END;
WHILE j > 0 DO
d := d DIV 2;
p := p DIV 2;
j := j-1;
IF r >= d THEN
r := r-d;
q := q+p
END
ENDThe first loop invariant is , , , and . Every doubling preserves it. For integer inputs, after at most iterations when ,
d > x; if , this loop has no iterations. The unbounded increase of proves termination. Equivalently, one can use the decreasing nonnegative loop variant while the guard is true.At the top of the second loop the loop invariant isIt holds on entry because
d > x = r. Suppose the old d is and the old p is . After the three halving assignments, and the old remainder satisfies . If , the desired new bound already holds. Otherwise subtracting leaves and adding to q preserves because . The divisions by two are exact: before each halving, j > 0 makes both p and d even.The loop variant
j decreases by one each iteration. At exit , so d = y and the loop invariant gives the required remainder bound. The work per iteration is constant in the unit-cost model, and the two loops have the same number of iterations. For ,The case , including , takes constant time. These are arithmetic-operation counts with sufficient range for the doubled values; arbitrary-precision arithmetic incurs additional bit complexity. If the inputs were arbitrary positive real numbers, the uniform bound would depend on rather than only ; the stated is the conventional integer division interpretation.Perform the elementary row operations from the bottom upwards:This order matters because each subtraction uses the predecessor's original row. For example,The other two differences are and . Thus the required upper triangular matrix isThese elementary row operations use no division, so the reduction remains valid for all real numbers , including .
For an matrix over the real numbers, write its rows as and its columns as . ThenThus the row space and column space consist of all linear combinations of the corresponding rows and columns; the null space is the kernel of a linear map represented by . The matrix rank counts independent row or column directions, with row rank equal to column rank.
If is positive and odd, is odd. The function is a strictly increasing function on the real numbers, so it is injective. Every real has the real odd root , so the function is also surjective.
If is positive and even, the exponent is positive and even. Then , while for all real . These facts disprove injectivity and surjectivity, respectively. ThereforeIf the convention is used, gives exponent zero. Interpreting as the constant polynomial makes the function neither injective nor surjective. If is left undefined, that case does not define the stated function on all the real numbers. Neither convention adds a qualifying value of .
Take the functionThen for every real number , while . Thus it is an involution different from the identity function.
Extend the data linked list node type by a real number field and compare that field through an Oberon type guard.Construct the data chain from Only data nodes reach
TYPE RealNode = RECORD (Node) value: REAL END;
RealTail = POINTER TO RealNode;
PROCEDURE RealLess(p, q: Tail): BOOLEAN;
BEGIN
RETURN p(RealTail).value < q(RealTail).value
END RealLess;RealTail nodes, retain the ordinary Header for its header, set its count correctly, and callMergeSort(RealLess, p)RealLess, so the header needs no real-valued field. Its precondition is that every data node has the RealNode extension. With ordinary real numbers, the comparison supplies a total order; machine values such as NaN must be excluded if the machine floating-point number type admits them.Represent the zero polynomial by ThusThe sentinel is a representation convention, not an assertion that the mathematical degree of the zero polynomial is an ordinary integer. In parts (b)–(e), algebraic correctness uses exact arithmetic on real numbers; ordinary floating-point arithmetic introduces rounding and needs an additional numerical policy if exact zero tests are to be replaced.
degree = -1, with an allocated one-element array whose element is zero. Its coefficient prefix is empty; the spare cell is merely convenient storage. For every nonzero polynomial, require degree >= 0, sufficient allocated capacity, and a nonzero leading polynomial coefficient.PROCEDURE Zero(VAR p: Poly);
BEGIN
p.degree := -1; NEW(p.coeff, 1); p.coeff[0] := 0.0
END Zero;
PROCEDURE Normalize(VAR p: Poly);
BEGIN
WHILE p.degree >= 0 DO
IF p.coeff[p.degree] # 0.0 THEN RETURN END;
DEC(p.degree)
END
END Normalize;Use polynomial convolution. If either input is the zero polynomial, return the zero polynomial immediately.Each pair of terms contributes , so the final polynomial coefficient of isWith exact arithmetic on real numbers, the product of the leading nonzero polynomial coefficients is nonzero, giving degree . The nested loops implement precisely this sum, preserve the input coefficient arrays, and take time for nonzero inputs.
PROCEDURE Mul(VAR prod: Poly; a, b: Poly);
VAR result: Poly; d, i, j: INTEGER;
BEGIN
IF (a.degree < 0) OR (b.degree < 0) THEN
Zero(result)
ELSE
d := a.degree+b.degree;
result.degree := d; NEW(result.coeff, d+1);
FOR i := 0 TO d DO result.coeff[i] := 0.0 END;
FOR i := 0 TO a.degree DO
FOR j := 0 TO b.degree DO
result.coeff[i+j] := result.coeff[i+j]
+ a.coeff[i]*b.coeff[j]
END
END;
Normalize(result)
END;
prod := result
END Mul;Scan the polynomial coefficients from highest degree downward. Omit zero terms, omit coefficient magnitude one on nonconstant terms, print The representation invariant ensures a nonzero polynomial prints at least one term. Constant and retain their coefficients, whereas and omit magnitude one. Thus the requested example printsDifferent
- for negative terms, and print + only before a positive term after the first. A helper prints small integral magnitudes without a decimal suffix; other real numbers use Out.Real.PROCEDURE OutMagnitude(c: REAL);
VAR n: INTEGER;
BEGIN
IF c < MAX(INTEGER) THEN
n := ENTIER(c);
IF c = n THEN Out.Int(n, 0) ELSE Out.Real(c, 0) END
ELSE
Out.Real(c, 0)
END
END OutMagnitude;
PROCEDURE OutPoly(p: Poly);
VAR i: INTEGER; c, magnitude: REAL; first: BOOLEAN;
BEGIN
IF p.degree < 0 THEN Out.String("0"); RETURN END;
first := TRUE; i := p.degree;
WHILE i >= 0 DO
c := p.coeff[i];
IF c # 0.0 THEN
IF c < 0.0 THEN Out.Char("-")
ELSIF ~first THEN Out.Char("+")
END;
magnitude := ABS(c);
IF (i = 0) OR (magnitude # 1.0) THEN
OutMagnitude(magnitude)
END;
IF i > 0 THEN
Out.Char("x");
IF i > 1 THEN Out.Char("^"); Out.Int(i, 0) END
END;
first := FALSE
END;
DEC(i)
END
END OutPoly;Out.Real implementations may use decimal or scientific notation for nonintegral coefficients; either denotes the appropriate real number. The number of coefficient inspections is linear in the stored polynomial degree.Use an immutable index and mutable value and links:For the sparse linked vector representation, a header
class Node(var value: Double, val index: Int,
var prev: Node = null, var next: Node = null)h has h.index >= 0, records that vector's dimension, and lies on a finite circular linked list. Every ordinary linked list node has an index in [0,h.index), a nonzero finite value, and a unique index; indices increase strictly from h.next to h.prev. For every reachable linked list node, including h, p.next.prev eq p and p.prev.next eq p. The abstraction function fills unspecified coordinates with zero. The header's value has no meaning. Finite real coordinates are assumed throughout; infinities and NaN are not ordinary real number coordinates for a mathematical inner product.Express existence of a minimum using only the given strict-order relation:In the closed interval , the witness is . In the open interval , every possible witness has the smaller member , so the first-order sentence is false. These witnesses are explanations in the ambient real numbers, not additional symbols in the first-order signature.
Use a
Double base so that the reciprocal is not truncated by integer division. Repeated multiplication gives the requested linear-work approximation to Euler's number.def eaLinear(n: Int): Double = {
require(n > 0)
val base = 1.0 + 1.0 / n.toDouble
var i = 0
var result = 1.0
while (i < n) {
result *= base
i += 1
}
result
}In exact arithmetic over the real numbers, the loop invariant is and . It holds initially, one multiplication establishes the next exponent, and at exit gives the specified power. The nonnegative loop variant is . There are exactly loop multiplications, hence in asymptotic analysis, and one division to form the base. With floating-point arithmetic, the invariant describes the intended exact result; the actual output has the corresponding rounding errors.
If is a skew-symmetric matrix, the scalar equals its own transpose, soand it is zero over the real numbers.
Conversely, suppose the quadratic form vanishes for every real vector. Substituting gives . Substituting gives for . Therefore . This proves the quadratic-form criterion for a skew-symmetric matrix:
The determinant is invariant under transpose, while multiplying all rows by multiplies it by . Thus, for odd integer ,Over the real numbers, this forces , so is a singular matrix. This is the odd-dimensional skew-symmetric singularity theorem.
Peak of a sequence 2026-10-25
A peak of a sequence of real numbers is an index whose value is at least every later value. This is a condition on the entire tail, stronger than comparing only adjacent terms. Equality is allowed: every index of a constant sequence is a peak of a sequence.
Infinitely many peaks of a sequence, listed in increasing order of index, give a nonincreasing monotone subsequence. If only finitely many exist, every index beyond the last has a larger later term; repeatedly choosing one gives a strictly increasing monotone subsequence. This proves the monotone subsequence theorem.
A peak of a sequence of supplies a lower bound for all subsequent terms of . If both and have a peak of a sequence, the two selected values bound a common tail. Adding bounds for the finite initial segment proves that is a bounded sequence.
Period group of a function 2026-10-14
Include zero when forming this set, even though a period of a function is normally required to be nonzero. For , is a subgroup of the additive group of the real numbers: composing two preserving shifts adds them, and reversing a preserving shift negates it. A constant function has period group . A nonconstant continuous real-valued function that is periodic has period group , where is its fundamental period.