What Is A Nonzero Digit

INTERMOLECULAR INTERACTIONS

In Ideas of Quantum Chemistry, 2007

Unmarried Exchange (SE) Mechanism

The smallest not-zero number of electron exchanges in  _ABC is equal to 1 (two electrons involved). Such an substitution may only take place betwixt two molecules, say, AB. ⁵⁶ . This results in terms of the order of S ⁱⁱ in the three-trunk expression. The third molecule does non participate in the electron exchanges, just is not just a spectator in the interaction (Fig. thirteen.11.a, b, c ). If it were, the interaction would not be three-torso.

Fig. thirteen.xi. A scheme of the SE and TE substitution not-additivities. Figs. (a), (b), (c) evidence the single exchange mechanism (SE). (a) Three non-interacting molecules (schematic representation of electron densities). (b) Pauli deformation of molecules A and B. (c) Electrostatic interaction of the Pauli deformation (resulting from substitution of electrons one and 2 between A and B) with the dipole moment of C. (d) The TE mechanism: molecules A and B exchange an electron with the arbitration of molecule C.

SE MECHANISM

Molecule C interacts electrostatically with the Pauli deformation of molecules A and B (i.e. with the multipoles that correspond the deformation). Such a mixed interaction is called the SE mechanism.

Information technology would be dainty to have a simple formula which could replace the tedious calculations involving the above equations. The three-body energy may be approximated ⁵⁷ by the product of the exponential term bexp(−aR_AB ) and the electrical field produced by C, calculated, east.m., in the middle of the altitude R_AB between molecules A and B. The goal of the exponential terms is to grasp the idea that the overlap integrals (and their squares) vanish exponentially with distance. The exponent a should depend on molecules A and B likewise as on their common orientation and reflects the hardness of both molecules. These kind of model formulae have low scientific value but are of practical use.

When the double electron exchanges are switched on, nosotros would obtain function of the three-body consequence of the order of S⁴. Since S is normally of the order of 10⁻², this contribution is expected to be small, although caution is brash, because the number of such terms is much more important.

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Infinite Words

In Pure and Applied Mathematics, 2004

Case 9.ane

Consider the automaton represented in Figure 9.5 , which recognizes the set of words having a finite nonzero number of b's. We detail the steps of Safra'south algorithm. The initial state is the tree with a single node of Figure 9.6. The action of the letters a and b on the initial state are represented in Effigy 9.7 and Figure 9.eight, respectively. A new state has now been created. The deportment of the letters a and b on this new state are represented in Effigy 9.ix and Figure 9.ten, respectively. Thus some other new land has been created. The action of the messages a and b on this new state are hands derived from the ones represented in Figures 9.nine and 9.10 by exchanging the names two and iii in every identify. After renaming u.s., we obtain the automaton of Figure ix.xi.

Figure 9.5. A Büchi automaton.

Figure 9.half-dozen. The initial land.

Figure 9.7. The action of a on the initial state.

Figure 9.eight. The activeness of b on the initial state.

Figure 9.nine. The activeness of a on the new state.

Effigy 9.10. The activeness of b on the new state.

Figure 9.11. The Rabin automaton obtained past Safra's algorithm.

We have L ₁ = ∅, L _two = {i, 3}, L _three = {1, two}, U _i = ∅, U ₂ = {two}, U _iii = {3}. Thus the accepting pairs are ({ane, iii},{2}) and ({i, 2}, {3}). Note that, by Formula (vii.4), these Rabin pairs are equivalent to the table T = {{2}, {3}}.

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Basic Representation Theory of Groups and Algebras

In Pure and Applied Mathematics, 1988

five.24

Now allow γ exist a fixed non-degenerate trace on A; let {kσ} be the not-cypher numbers determining γ by (13); and define a bilinear "inner product" on A:

$\begin{array}{cc}\left[a,b\right]=\gamma \left(ab\right)& \left(a,b\in A\right).\end{array}$

In view of the non-degeneracy of γ, we can employ [,] to place A and its adjoint infinite A^# in one-to-ane correspondence. If α ∈ A^# , let α ^∼ exist the corresponding element of A, given by

$\begin{array}{cc}\left[a^{\sim },b\right]=\alpha \left(b\right)& \left(b\in A\right).\end{array}$

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NATURAL NUMBERS

Herbert B. Enderton , in Elements of Gear up Theory, 1977

Exercises

18.

Simplify: ${\in }_{\omega }^{-1}〚\{\mathrm{vii},8\}〛$ .

19.

Testify that if m is a natural number and d is a nonzero number, then there exist numbers q and r such that thou = (d · q) + r and r is less than d.

20.

Let A be a nonempty subset of ω such that ∪A = A. Show that A = ω.

21.

Bear witness that no natural number is a subset of any of its elements.

22.

Evidence that for whatever natural numbers chiliad and p nosotros have g ∈ m + p +.

23.

Assume that m and northward are natural numbers with m less than north. Prove that there is some p in ω for which grand + p ⁺ = n. (Information technology follows from this and the preceding exercise that m is less than north iff (∃p ∈ ω) m + p ⁺ = n.)

24.

Assume that m + north = p + q. Show that

$\begin{array}{ccc}m\in p& \iff & q\in \mathrm{north}\end{array}.$

25.

Assume that n ∈ yard and q ∈ p. Bear witness that

$(\mathrm{one\; thousand}\cdot q)+(n\cdot p)\in (m\cdot p)+(n\cdot q).$

[Suggestion: Use Exercise 23.]

26.

Assume that north ∈ ω and f : due north ⁺ → ω. Evidence that ran f has a largest chemical element.

27.

Presume that A is a set, G is a function, and f _i and f _two map ω into A. Further presume that for each n in ω both

and

belong to dom G and

Show that f ₁ = f ₂.

28.

Rewrite the proof of Theorem 4G using, in place of induction, the wellordering of ω.

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Representation of Numbers past Decomposable Forms

In Pure and Practical Mathematics, 1966

Lemma three.

Let K exist an algebraic number field of degree n = s + 2t and let K exist a total module in K with discriminant D . Then in that location exists a nonzero number α in M whose norm satisfies

(6.5) $\left|N\left(\alpha \right)\right|\le {\left(\frac{\mathrm{two}}{\pi }\right)}^t\sqrt{\left|D\right|}\cdot$

Proof. We take positive existent numbers c ₁,…, c_south+t , so that

(6.six) $c_1\cdots c_{\mathrm{south}+t}={\left(\frac{2}{\pi }\right)}^t\sqrt{\left|D\right|}+\varepsilon ,$

where ε is an capricious positive real number. From Theorems 2 and iv of Section 4 information technology follows that there exists a number α ≠ θ in M satisfying

$\begin{array}{cccc}\left|{\sigma }_k\left(\alpha \right)\right|<c_k& \left(\mathrm{one}\le k\le s\right),& {\left|{\sigma }_{s+j}\left(\alpha \right)\right|}^{\mathrm{two}}<c_{s+j}& \left(1\le j\le t\right)\cdot \end{array}$

The norm

$\mathrm{Due\; north}\left(\alpha \right)={\sigma }_1\left(\alpha \right)\cdots {\sigma }_s\left(\alpha \right){\left|{\sigma }_{\mathrm{due\; south}+1}\left(\alpha \right)\right|}^2\cdots {\left|{\sigma }_{s+\mathrm{one}}\left(\alpha \right)\right|}^2$

of such a number must take absolute value not exceeding (6.6). Since this is true for arbitrarily small ε, in that location must be a nonzero number of M which satisfies the inequality (half dozen.five).

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Equivalent Representations of Bent Functions

Natalia Tokareva , in Bent Functions, 2015

half dozen.5 Sets of Subspaces

Let us consider a geometrical representation for the grade of aptitude functions introduced past Carlet and Guillot [61] in 1998 (encounter as well an earlier paper [lx]).

Permit f be a Boolean function in due north variables. Let ${\text{Ind}}_{\mathrm{South}}:{\mathbb{F}}_2^n\to {\mathbb{F}}_2$ exist the characteristic function of the subset $S\subseteq {\mathbb{F}}_2^n$ —that is, Ind _S takes the value ane on elements from Southward and the value 0 on all other elements.

Theorem 29

A function f is aptitude if and only if at that place are subspaces East _one,…,E _grand of dimension n/2 or (north/2) + ane in the space ${\mathbb{F}}_2^n$ and nonzero numbers m ₁,…,k _k such that for whatever $x\in {\mathbb{F}}_2^n$

$\sum _{i=\mathrm{i}}^k{m}_i{\mathrm{Ind}}_{E_i}(\mathrm{10})=\pm 2^{(\mathrm{north}/\mathrm{two})-1}{\mathrm{Ind}}_{\{0\}}(x)+f(\mathrm{ten}).$

Carlet and Guillot [61] proposed restrictions on the pick of subspaces E ₁,…,E ₁₀₀₀ , such that this choice becomes unique for every bent office. Thus, information technology is possible to make this representation unique for every aptitude function.

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Distances Between Aptitude Functions

Natalia Tokareva , in Aptitude Functions, 2015

11.2 Classification of Bent Functions at the Minimal Distance from the Quadratic Bent Role

It is well known that extended affine equivalence preserves Hamming distances between Boolean functions—that is,

$\mathrm{dist}(f,g)=\mathrm{dist}(A(f),A(\mathrm{grand})),$

where A(⋅) is an arbitrary nondegenerate affine transformation of a Boolean office (i.e., a nondegenerate affine transformation of variables and calculation an affine Boolean office; see Section 5.2).

According to Theorem 18, whatsoever quadratic aptitude function in n variables is equivalent to the function q(x) = 10 _i ten _(n/ii)+1 ⊕ 10 ₂ x _(n/ii)+2 ⊕⋯ ⊕ x _{northward/two} x _north . Information technology is user-friendly for u.s.a. to bargain with a quadratic bent role of this form. Thus, information technology is interesting to study bent functions at the minimal distance from only 1 aptitude function q.

To classify all aptitude functions at altitude ii ^n/2 from q let us innovate some annotation. Permit M be a binary 1000 × n matrix in reduced row echelon form without nada rows. This ways that

•

the leading number (the showtime nonzero number from the left) of a nonzero row is always strictly to the correct of the leading coefficient of the row above it;

•

all entries in a column higher up a leading entry are zeroes;

•

in that location are exactly k leading numbers.

Such a matrix is called also a Gaussian-Hashemite kingdom of jordan matrix (without nil rows). So, Chiliad looks like

$\mathrm{Thou}=\left(\begin{array}{llllllll}\mathrm{i}& *& 0& 0& *& 0& *& \dots \\ 0& 0& 1& 0& *& 0& *& \dots \\ 0& 0& 0& 1& *& 0& *& \dots \\ 0& 0& 0& 0& 0& \mathrm{ane}& *& \dots \\ \dots & \dots & \dots & \dots \\ \end{array}\right).$

Annotation that for every linear subspace $L\subseteq {\mathbb{F}}_2^{\mathrm{due\; north}}$ of dimension yard there is a unique matrix M in the reduced row echelon form with rows being basic vectors. So the matrix is usually chosen the Gaussian-Jordan matrix of a subspace L.

Denote by ℓ(M) the subset of positions of leading numbers of rows in a matrix M. For instance, in our instance given above, ℓ(M) = {1,3,4,6,…}.

In full general, let the Gaussian-Hashemite kingdom of jordan matrix have the form

Let united states phone call a Gaussian-Jordan n/2 × n matrix Madmissible of order t if the following atmospheric condition hold:

•

A is a Gaussian-Hashemite kingdom of jordan t × north/two matrix.

•

C is a Gaussian-Jordan (n/ii − t) × n/two matrix.

•

Linear subspaces of ${\mathbb{F}}_{\mathrm{two}}^{\mathrm{northward}/\mathrm{ii}}$ with Gaussian-Jordan matrices A and C are orthogonal—that is, A ⋅ C ^T is the zero matrix.

•

All columns of B with numbers from ℓ(C) are aught.

•

A′ and B′ are matrices of size t × t obtained from A and B, respectively, after elimination of columns with numbers from ℓ(C). Then if t > one,

$\left(\begin{array}{llllll}\hfill {a{}^{\prime }}^{(2)}\hfill & \hfill {a{}^{\prime }}^{(\mathrm{ane})}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill \dots \hfill \\ \hfill {a{}^{\prime }}^{(t)}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill {a{}^{\prime }}^{(1)}\hfill \\ \hfill \dots \hfill \\ \hfill 0\hfill & \hfill {a{}^{\prime }}^{(\mathrm{iii})}\hfill & \hfill {a{}^{\prime }}^{(2)}\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill \dots \hfill \\ \hfill 0\hfill & \hfill {a{}^{\prime }}^{(t)}\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill {a{}^{\prime }}^{(2)}\hfill \\ \hfill \dots \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill {a{}^{\prime }}^{(t)}\hfill & \hfill {a{}^{\prime }}^{(t-\mathrm{ane})}\hfill \\ \hfill \hfill \end{array}\right)\cdot \left(\begin{array}{l}\hfill {{b{}^{\prime }}^{(1)}}^{\mathrm{T}}\hfill \\ \hfill {{b{}^{\prime }}^{(2)}}^{\mathrm{T}}\hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill ⋮\hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill {{b{}^{\prime }}^{(t)}}^{\mathrm{T}}\hfill \\ \hfill \hfill \end{array}\right)=0,$

where a′ ⁽ⁱ⁾ and b′ ⁽ⁱ⁾ are the ithursday rows of A′ and B′, respectively.

Kolomeec proved [208] the following theorem (see also [210]):

Theorem 75

Let V be an affine subspace of ${\mathbb{F}}_2^n$ of dimension n/2. Then the bent function q(10) = x ₁ x _{(northward/2)+1} ⊕ x _ii x _(n/ii)+2 ⊕⋯ ⊕ x _n/2 x _n is affine on 5 if and but if V is a shift of a linear subspace with an admissible Gaussian-Jordan matrix.

For example, if north = 4, the bent function x ₁ 10 _ii ⊕ 10 ₃ 10 ₄ is affine on all linear subspaces (and all their shifts) with Gaussian-Hashemite kingdom of jordan matrices

$\left(\begin{array}{llll}0& 0& 1& 0\\ 0& 0& 0& 1\\ \end{array}\right),\left(\begin{array}{llll}1& 0& *& 0\\ 0& 0& 0& \mathrm{ane}\\ \end{array}\right),\left(\begin{array}{llll}\mathrm{ane}& 1& 0& *\\ 0& 0& 1& \mathrm{ane}\\ \end{array}\right),$

$\left(\begin{array}{llll}0& \mathrm{one}& 0& *\\ 0& 0& \mathrm{i}& 0\\ \end{array}\right),\left(\begin{array}{llll}\mathrm{ane}& 0& *& 0\\ 0& 1& 0& *\\ \end{array}\right),\left(\begin{array}{llll}\mathrm{one}& 0& *& 1\\ 0& \mathrm{ane}& 1& *\\ \end{array}\right).$

Thus, by Theorems 74 and 75 in order to count the number of bent functions at a altitude 2 ^northward/2 from the quadratic aptitude function 1 has to enumerate all the affine subspaces of dimension n/2 with open-door Gaussian-Jordan matrices. This is done in [208], where the following theorem is proven:

Theorem 76

Any quadratic aptitude office in n variables has exactly $2^{\mathrm{due\; north}/2}{\prod }_{i=1}^{n/2}(2^i+\mathrm{one})$ bent functions at the minimal possible distance 2 ^n/2 from it.

For instance, there are 60 bent functions at the minimal possible altitude 4 from a bent function in iv variables (since every such bent function is quadratic).

The number from Theorem 76 tin be estimated every bit

$2^{n(\mathrm{north}+6)/8}<2^{n/2}\cdot (2^{\mathrm{ane}}+1)\cdot \cdots \cdot (2^{n/2}+\mathrm{i})<\mathrm{iii}\cdot 2^{\mathrm{due\; north}(n+\mathrm{half\; dozen})/8}.$

Note that asymptotically

$2^{n/2}\cdot (2^1+\mathrm{one})\cdot \cdots \cdot (2^{n/\mathrm{two}}+1)\approx 2.38\cdot {\mathrm{ii}}^{n(n+\mathrm{half-dozen})/8}.$

It is interesting that more one-third of open-door Gaussian-Jordan matrices (that we use in Theorem 75) can be synthetic in a very simple style. Namely, all matrices of the grade $\left(\begin{array}{ll}\mathrm{Due\; east}& T\\ \end{array}\right)$ , where E is the identity matrix of size due north/two × n/two and matrix T is an capricious symmetric matrix of size northward/2 × due north/ii, are admissible Gaussian-Jordan matrices.

Note also that every bent office at the minimal distance 2 ^northward/2 from the quadratic bent part in north variables is equivalent to a McFarland bent part.

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Generalized Kac-Moody algebras

N. Sthanumoorthy , in Introduction to Finite and Infinite Dimensional Lie (Super)algebras, 2016

Complete classification of special imaginary roots of BKM algebras possessing infinite number of elementary imaginary roots

For GKM algebras corresponding to BCMs of space order, we have the following cases:

(i)

All uncomplicated roots are imaginary: These are GKM algebras whose BCMs practise not appear as extensions of generalized Cartan matrices of Kac-Moody algebras. Then if we leave the corresponding rows and columns containing negative diagonal elements of BCMs, nosotros will non go generalized Cartan matrices of Kac-Moody algebras. Hence Weyl group in each case is empty because these algebras do not comprise elementary real roots. So there is no special imaginary root.

(ii)

Finite (nonzero) number of simple existent roots and infinite number of uncomplicated imaginary roots: Hither infinite number of negative elements will occur anywhere along the diagonal of BCMs. In this case, leaving the rows and columns containing these negative diagonal elements, we will go the generalized Cartan matrices of the Kac-Moody algebras. This may be finite, affine, or indefinite type. Using the definition of special imaginary roots, i can easily conclude that if α is a special imaginary root, the equations corresponding to Weyl reflections r _α cannot be solved. And then GKM algebras with space number of simple imaginary roots and finite (nonzero) number of existent simple roots do not possess special imaginary roots.

For example, consider the Monster Lie algebra. This algebra has 1 real elementary root α _ane and infinite number of simple imaginary roots α _i for $2\le i\le \infty .$ Using the definition of special imaginary roots, one can hands conclude that if α is a special imaginary root, the equations respective to Weyl reflections r _α cannot be solved. The coefficients 50 _i of ${\alpha }_i(2\le i\le \infty )$ in the equations respective to Weyl reflections r _α of dissimilar simple imaginary roots α _i cannot satisfy simultaneously all the equations. Hence Monster Lie algebra does not possess special imaginary roots.

(three)

Space number of real uncomplicated roots and finite number of imaginary elementary roots: As the number of real simple roots is infinite, the order of the Weyl group is space. So if α is a special imaginary root, ane cannot verify simultaneously space number of equations for α to satisfy and hence we conclude that at that place exists no special imaginary root.

(four)

All simple roots are real: Here the Weyl group has infinite number of elements and there is no imaginary simple root. Hence we conclude that there exists no special imaginary root.

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Algebraic Geometry

Rick Miranda , in Encyclopedia of Concrete Science and Technology (Tertiary Edition), 2003

I.A Affine Infinite

To be more precise about solutions to polynomial equations, one first specifies a field 1000 of numbers where the coefficients of the polynomials prevarication, and where one looks for solutions. A field is a set with two binary operations (add-on and multiplication) in which all of the usual rules of arithmetic agree, including subtraction and division (by nonzero numbers). The chief examples of involvement are the field ℚ of rational numbers, the field ℝ of real numbers, and the field ℁ of complex numbers. Another is the finite field ℤ/ p of the integers {0,1,   …,p–ane} under addition and multiplication modulo a prime number p.

Solutions to polynomials in n variables would naturally be northward-tuples of elements of the field. If we denote the field in question by Chiliad, the natural place to find solutions is affine n-space over K, denoted by G ⁿ , A ⁿ _K , or simply A ^{due north} :

$A^{\text{n}}=\{\underset{―}{\mathbf{z}}=({\text{z}}_1,{\text{z}}_2,\dots ,{\text{z}}_{\text{n}})|{\text{z}}_{\text{i}}\in \text{K}\}.$

When n  =   1 we have the affine line, if n  =   2 we have the affine plane, and so along.

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Computational Error and Complexity in Science and Engineering

V. Lakshmikantham , S.K. Sen , in Mathematics in Science and Engineering, 2005

iv.nine.7 Role of NaN

Division by 0, 0 divided by 0, square-root of a negative number are operations which cause unrecoverable error and consequently traditionally halt the computation. This halting is not desirable in several situations. Consider, for example, a stock-still point iteration scheme to compute roots of an equation in ane or more variables. It might so happen that for 1 of the several roots, the scheme could come up beyond 0/0 or a sectionalization of a finite nonzero number by 0 or square-root of a negative number. In such a situation, the computation would halt undesirably aborting the root-finding process. Consider the equation $x+{\displaystyle \surd }({\text{x}}^{\mathrm{ii}}-\mathrm{three}\text{ten})-\mathrm{i}=0$ which has one root at x = −1. If we apply the fixed betoken iteration scheme

${\text{x}}_{\text{i}+1}=\mathrm{one}-\surd \left({\text{x}}_{\text{i}}^2-\mathrm{iii}{\text{ten}}_{\text{i}}\right),\text{i}=0,1,\dots ,\left|{\text{x}}_{\text{i}+\mathrm{i}}-{\text{x}}_{\text{i}}\right|/\left|{\text{x}}_{\text{i}+1}\right|\le .5\times {\mathrm{x}}^{-4},$

where x₀ is an initial approximation. If we cull x₀ = 1 then the computation of ten_i would encounter the computation of the foursquare-root of −2 — an unrecoverable error. Consequently the root-finding scheme will be aborted/halted. If we at present consider the equation (x⁴ − 1)(10 + √(ten² − 3x) − ane) = 0 which has x = 1 and x = −1 every bit two of the roots and use the scheme

$\begin{array}{c}{\text{x}}_{\text{i}+\mathrm{ane}}={\text{x}}_{\text{i}}^4\left({\text{x}}_{\text{i}}+\surd \left({\text{x}}_{\text{i}}^2-\mathrm{three}{\text{ten}}_{\text{i}}\right)-\mathrm{ane}\right)+\mathrm{i}-\surd \left({\text{x}}_{\text{i}}^{\mathrm{ii}}-\mathrm{iii}{\text{x}}_{\text{i}}\right),\\ \text{i}=0,\mathrm{one},\dots ,\left|{\text{ten}}_{\text{i}+1}-{\text{10}}_{\text{i}}\right|/\left|{\text{x}}_{\text{i}+1}\right|\le .5\times {10}^{-4},\end{array}$

where ten₀ is an initial approximation. If we choose x₀ = 1 then, as before, the computation of 10_i would come across the ciphering of the square-root of −two — an unrecoverable error although x = 1 is a root. The trouble of halting tin can be obviated by introducing NaN that will exist produced in the arithmetic operations ∝ + (−∝), 0 × ∝, 0/0, ∝/∝, ∝ rem x, x rem 0, and ${\displaystyle \surd }(-\mathrm{ten})$ , where 10 is a valid positive finite floating point number. When roots of an equation f(x) = 0 are attempted to be computed exterior the domain of f due to maybe some wrong initial approximation (guess), the execution of the subprogram for f will return a NaN without stopping the root-finding process. If a NaN is an operand in a floating indicate operation, then the output volition be a NaN. If the value of a subexpression is a NaN, then the value of the entire expression volition be a NaN.

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What Is A Nonzero Digit,

Source: https://www.sciencedirect.com/topics/mathematics/nonzero-number

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