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* A. R. Rajwade, ''Squares'', London Mathematical Society Lecture Note Series '''171''' Cambridge University Press (1993) {{ISBN|0-521-42668-5}} {

Note that the geometry of special relativity theory (cf. [[Space-time|Space-time

Note that a lower bound for the first eigenvalue, without any curvature assumpti ...</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> P. Buser, "A note on the isoperimetric constant" ''Ann. Sci. Ecole Norm. Sup.'' , '''15'''

<TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Barbier, "Note sur le problème de l'ainguille et le jeu du joint couvert" ''J. Math. Pur

Note that this norm differs from the [[operator norm]] of $A$ (for instance beca

...liations on surfaces) are equivalent (cf. also [[Lamination|Lamination]]). Note also that the construction shows that the (weighted) train track is in a se ...nd laminations in higher-dimensional manifolds (see [[#References|[a2]]]). Note that a notion close to the notion of train track is already contained in [[

* A. R. Rajwade, ''Squares'', London Mathematical Society Lecture Note Series '''171''' Cambridge University Press (1993) {{ISBN|0-521-42668-5}} {

...<TD valign="top">[a4]</TD> <TD valign="top"> E.A. Michael, "Yet another note on paracompact spaces" ''Proc. Amer. Math. Soc.'' , '''10''' (1959) pp.

Note that $A_5$ is the non-Abelian simple group of smallest possible order.

...to algebraic geometry and topology'', London Mathematical Society Lecture Note Series '''217''' Cambridge University Press (1995) {{ISBN|0-521-46755-1}} { * A. R. Rajwade, ''Squares'', London Mathematical Society Lecture Note Series '''171''' Cambridge University Press (1993) {{ISBN|0-521-42668-5}} {

Note that the best coordinates are neither the normal nor the harmonic ones, but

...properties. This Ray resolvent is associated to a semi-group $(\hat P_t)$ (note that $\hat P_0$ need not be the identity: existence of branching points), q

.... H. Greaves, G. Harman, M. N. Huxley; London Mathematical Society Lecture Note Series '''237''', Cambridge University Press (1997) {{ISBN|0-521-58957-6}}

...</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R.R. Hall, "A note on Farey series" ''J. London Math. Soc.'' , '''2''' (1970) pp. 139–148

...On graphs with least eigenvalue $−2$'' London Mathematical Society Lecture Note Series '''314''' Cambridge University Press(2004) {{ISBN|0-521-83663-8}} {{

...ce $ X $ onto a topological space $ B $ (i.e., a [[Fibration|fibration]]). Note that $ X $, $ B $ and $ \pi $ are also called the '''total space''', the '' ...f(b) \stackrel{\text{df}}{=} (\pi \circ F)[{\pi^{\leftarrow}}[\{ b \}]] $. Note that $ F $ is a [[Covering|covering]] of $ f $ and that $ \pi_{1} \circ F =

...en formal system is decidable, then such a system is said to be complete. (Note that it is impossible to require that all, and not just the closed, formula

* A. R. Rajwade, ''Squares'', London Mathematical Society Lecture Note Series '''171''' Cambridge University Press (1993) {{ISBN|0-521-42668-5}} {

...$(A_1\mathbin{\&}\dotsb\mathbin{\&}A_n)\supseteq(B_1\lor\dotsb\lor B_m)$ (note that an empty conjunction denotes truth, and an empty disjunction denotes f

...f an already defined population of objects under study. It is essential to note that the condition which defines the species is to be understood in the int

This result had been conjectured by M.A. Lavrent'ev in 1938. Note that the exponential function shows that there is no such result for $n=2$.

* Neumann, B.H. ''A note on algebraically closed groups'' ''J. Lond. Math. Soc.'' '''27''' (1952) 24

...1]</TD> <TD valign="top"> B. Jessen, J. Marcinkiewicz, A. Zygmund, "Note on the differentiability of multiple integrals" ''Fund. Math.'' , '''25'''

...oof of the expression for $S_n$ see, e.g., [[#References|[a1]]], Thm. 422. Note that the series $\sum 1/p$ extended over all prime numbers $p$ diverges als

Note that the same point in the boundary of $ D $

Note that, when $B (t)$ is constant, \eqref{e:conclusion2} coincides with \eqref |valign="top"|{{Ref|Gr}}|| T. H. Gronwall, "Note on the derivatives with respect to a parameter of the solutions of a system

..., then the series converges if $\alpha> 1$ and diverges if $\alpha\leq 1$. Note that for \eqref{e:Gauss} to be valid it is necessary (but not sufficient) t

Note that a Lebesgue point is, therefore, a point where $f$ is [[Approximate con

...M. Cohn, ''Skew Field Constructions'', London Mathematical Society lecture note series '''27''', Cambridge University Press (1977) {{ISBN|0-521-21497-1}}

...l); in $P_1 * P_2$, then all such pairs have $p_1 \prec p_2$ (series). We note that $+$ is commutative but $*$ is not.

...is, by definition, the set of morphisms in $\mathfrak{K}$ from $X$ to $T$. Note that any two final objects of a category are (canonically) isomorphic, and

===Historical note===

...TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> N. Jacobson, "A note on automorphisms and derivations of Lie algebras" ''Proc. Amer. Math. Soc.

...TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> I. Stavrov, "A note on generalized Osserman manifolds" ''Preprint'' (1998)</TD></TR></table>

Note that it is then an eigen value of the integral operator defined by $ K( x

Note that for $ W $

<TR><TD valign="top">[a2]</TD> <TD valign="top"> Z. Jelonek, "A note about the extension of polynomial embeddings" ''Bull. Polon. Acad. Sci. Ma

.../TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J.O. Hennefeld, "A note on the Arens products" ''Pacific J. Math.'' , '''26''' (1968) pp. 115– <TR><TD valign="top">[a9]</TD> <TD valign="top"> Á. Rodriguez-Palacios, "A note on Arens regularity" ''Quart. J. Math. Oxford Ser. (2)'' , '''38''' : 149

Cf. also [[Differential invariant|Differential invariant]]. Note however that for differential invariants not only tensor bundles but also (

...4)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Okubo, "Note on unitary symmetry in strong interactions" ''Progress Theor. Phys.'' , ''

<TR><TD valign="top">[a2]</TD> <TD valign="top"> H.J.J. te Riele, "A Note on the Catalan–Dickson Conjecture" , ''Mathematics of Computation'' '''27

...amental group|fundamental group]] $\pi_1 (\Omega)$ of $\Omega$ is trivial. Note that the connectedness assumptions guarantees that, if $\gamma$ can be defo

...ussian for $d \geq 3$ and non-Gaussian for $d = 2$ ([[#References|[a7]]]). Note that for $d \geq 2$ the Wiener sausage is a sparse object: since the Browni to leading order. Note that, apparently, a large deviation below the scale of the mean "squeezes

<TR><TD valign="top">[3]</TD> <TD valign="top"> L. Gårding, "Note on continuous representations of Lie groups" ''Proc. Nat. Acad. Sci. USA''

...e analogy with the classical definition of probability is evident. One may note that in the [[Bertrand paradox|Bertrand paradox]], which is connected with

...<TD valign="top">[a1]</TD> <TD valign="top"> E.M. Alfsen, P. Holm, "A note on compact representations and almost periodicity in topological groups" '

is well defined. Note that the real part of $ T $

<TR><TD valign="top">[a2]</TD> <TD valign="top"> R.D. Carmichael, "Note on a new number theory function" ''Bull. Amer. Math. Soc.'' , '''16''' (1

<TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Leech, "Note on sphere packings" ''Canad. J. Math.'' , '''19''' (1967) pp. 251–267<

...n coverings (cf. [[Nerve of a family of sets|Nerve of a family of sets]]). Note that a geometric complex is also a cellular complex (cf. [[Cell complex|Cel

Note that if $\varepsilon = - 1$ and $K ( a , b ) \equiv 0$ (identically), then Note that a "triple system" in the sense discussed above is totally different

* {{Ref|a1}} E. Catalan, "Note sur une équation aux différences finies" ''J. Math. Pures Appl.'' , '''3

Note that this definition, which is commonly used now (see [[#References|[a1]]],

...nts of this relation process these words to yield the same result. One may note that if some binary relation is decidable (cf. [[Decidable set|Decidable se

Note that, as an obvious corollary, if the interval $I$ has length larger than $

...rsal algebra, and by the set of all closed subsets in a topological space (note that while the meet of a family of closed sets is their set-theoretic inter

* A. R. Rajwade, ''Squares'', London Mathematical Society Lecture Note Series '''171''' Cambridge University Press (1993) {{ISBN|0-521-42668-5}}{{

...psilon(m) = 1$ for all $m$ and its inverse $\mu$, the [[Möbius function]]. Note that $\phi = \mu * N_1$, where $N_1(n) = n$ for all $n$, and that $\tau = \

...d></tr><tr><td valign="top">[12]</td> <td valign="top"> B. Kamiński, "A note on $K$-systems" ''Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys.'' ,

...End}(P,{+})$ be the ring of endomorphisms of the additive group of $P$ and note that $\text{End}(P,{+})$ is a left [[module]] over $P$. There is a one-to-o

<TR><TD valign="top">[5]</TD> <TD valign="top"> Y. Nakai, "Note on the intersection of an algebraic variety with the generic hyperplane" ''

...quivalent (over $\Sigma ^ { * }$) if they have exactly the same solutions. Note that only systems of equations without constants are considered here. Howev ...]] and rings of integer $( 2 \times 2 )$-matrices in [[#References|[a5]]]. Note that in the latter proof an algebraic structure with two operations, namely

from physics rather than economics or sociology", as Fortet puts it in his note interesting. Let us note particularly his work on the

...[a3]</TD> <TD valign="top"> S. Grandlund, P. Lindqvist, O. Martio, "Note on the PWB-method in the non-linear case" ''Pacific J. Math.'' , '''125'''

Note that a triple system in this sense is completely different from, e.g., the

...</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Mazur, "A note on the growth of Davenport's constant" ''Manuscr. Math.'' , '''74''' (199

...with weights $w _ { 1 } = ( 1 + c ) / 2$ and $w _ { 2 } = ( 1 - c ) / 2$. Note that if the weights are given by $w _ { 1 } = ( 1 + \operatorname { sign } ...reversing the order of differencing in the predictor and corrector steps. Note that the MacCormack scheme (a11) is not written in conservative form (a2).

...gy principal series instead of non-degenerate continuous principal series. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.

...<TR><TD valign="top">[2]</TD> <TD valign="top"> D. Gale, O. Gross, "A note on polynomial and separable games" ''Pacific J. Math.'' , '''8''' : 4 (1

...linear, then $x=\mathsf{E}(X|Y=y)=\mu_x+\rho(\sigma_x/\sigma_y)(y-\mu_y)$. Note that the product of the two slopes is $\rho^2$.

...complicated. For an elegant and concise proof see 2.10.43 of {{Cite|Fe}}. Note that a Lipschitz extension with a non-optimal constant can be easily achiev

...countable collection $\{E_i\}$ of ''disjoint'' elements of $\mathcal{S}$. Note that, since we assume $\mu (\emptyset) = 0$, a measure is always finitely a

...ub-formulas. Consider, for instance, a formula $P$ with probability $0.5$. Note that its negation, $\neg P$, will also have probability $0.5$. Now one woul

Note that the random effects in \ref{lin mix eff model 1} implicitly imply $\Sigma_i V_i^{-1}$ and $I_{n_i} - \Sigma_i V_i^{-1}$, respectively. Note that

...fect if $\sigma ( n ) = 2 n$ (cf. also [[Perfect number|Perfect number]]). Note that some authors call a number $n$ abundant if $\sigma ( n ) \geq 2 n$. Cl ...d></tr><tr><td valign="top">[a20]</td> <td valign="top"> H.N. Shapiro, "Note on a theorem of Dickson" ''Bull. Amer. Math. Soc.'' , '''55''' (1949) pp

...point in each direction at least $| D ( C ) |$ times. It is interesting to note that this is true even for differentiable locally one-to-one curves which a

...usual "tensor products" give monoidal categories — whence the notation. Note that one cannot identify all isomorphic objects in $\mathcal{C}$.

...relations between them, complex functioning, hierarchy of structure, etc. Note that the terms complex system and "large (scale) system" are not synonymo

...that $\mathbf{x} ^ { 0 }$ is kept constant in the differentiation of $f$. Note that, like $f$, $D f / D t$ is a function of $\mathbf{x} ^ { 0 }$ and $t$ b ...(a10) serves as a formula for the material derivative in the spatial form. Note that the first term on the right-hand side of this formula, namely $\partia

...</td></tr><tr><td valign="top">[a11]</td> <td valign="top"> R. Rado, "A note on Bernoullian numbers" ''J. London Math. Soc.'' , '''9''' (1934) pp. 88

Note that the $ \gamma _ {jk } ^ {i} $ by the [[Euler theorem|Euler theorem]] on homogeneous functions. Note that $ G ^ {i} $,

...of $\phi$ and $\operatorname{conv} ( E )$ is the convex hull of a set $E$. Note that (a1) is a combination of an improvement of a Hartman–Wintner theorem Note that this theorem is equivalent to the Helson–Szegö theorem on weighted

Note that if $G$ is an $M$-group and if $q$ is a unit of $B ( G )$, then conjuga ...} \neq 0$. Of course, $\sigma$ is $X$-outer when $\Phi _ { \sigma } = 0$. Note that, in the case of semi-prime rings, $\operatorname{Inn} ( R )$ need not

...a8]</TD> <TD valign="top"> E.M. Wright, "Burnside's lemma: a historical note" ''J. Combin. Th. B'' , '''30''' (1981) pp. 89–90</TD></TR>

<TR><TD valign="top">[2]</TD> <TD valign="top"> G.H. Hardy, "Note on Ramanujan's trigonometrical function $c_q(n)$ and certain series of arit

Note that this kind of triple system is completely different from the combinator

In his 1849 note, Listing showed that the figure-eight knot, Fig.a1b, and its mirror image a

It is important to note that the $\Gamma^k_{ij}$ are not the components of a tensor field. In fact

Note that as the Poisson density is given by $f(y_i|x_i)=\frac{1}{y_i!}\exp[y_il $\mbox{var}[Y_{it}]=\sigma_{itt}.$ Note that both $\mu_{it}$ and $\sigma_{itt}$ are functions of $\beta.$ But, when

...u_n = a \right\}$ and $\left\{0 \le n \le 2^{k+1} - 1 : u_n = b \right\}$. Note that this partition is conjectured to be the unique partition of the set $\ [[unavoidable pattern]]s in strings. Note that such combinatorial properties were used to solve various algebraic pro

...product of the fibres $\a^{-1}(c)\subseteq A$ and $\b^{-1}(c)\subseteq B$. Note also that (binary) products (cf.

<TR><TD valign="top">[a1]</TD> <TD valign="top"> K.A.H. Gravett, "Note on a result of Krull" ''Proc. Cambridge Philos. Soc.'' , '''52''' (1956) pp

* ''In case you encounter any problems with the new software just drop a note on the [[Talk:Main_Page | discussion page]] of this page.''

<tr><td valign="top">[a1]</td> <td valign="top"> F. Beukers, "A note on the irrationality of $\zeta ( 3 )$" ''Bull. London Math. Soc.'' , '''11

Note that each point is incident with 2 lines, each line is incident with 2 poin

...TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.B. Cruse, "A note on one-factors in certain regular multigraphs" ''Discrete Math.'' , '''18'

Note that in some of the literature dealing with webs, nets and pencils in the s (Note that a square matrix $ P ( \lambda ) $

...pposed to the $n$ Gershgorin discs in (a2). But it is equally important to note that the eigenvalue inclusions of (a2) and (a5) use the exact same data fro

'''Note:''' if you decided not to use the folder ''Documents/EOM_Scripts'', you ne '''Note:''' the It's All Text add-on saves temporary files in <nowiki>Library/Appli

...td></tr><tr><td valign="top">[a6]</td> <td valign="top"> P. Traczyk, "A note on rotant links" ''J. Knot Th. Ramifications'' , '''8''' : 3 (1999) pp.

...TR><TD valign="top">[1]</TD> <TD valign="top"> H. Nikaido, K. Isoda, "Note on non-cooperative convex games" ''Pacific J. Math.'' , '''5''' (1955) p

...align="top">[a10]</TD> <TD valign="top"> C.D. Savage, C.-Q. Zhang, "A note on the connectivity of acyclic orientation graphs" ''Discrete Math.'' (to

...rily disconnected (because they have no non-trivial connected subspaces). (Note that ii) implies i).)

See [[#References|[a1]]] for a well-known additional reference. Note that the Hilbert space obtained by completion of the set of all $ C ^ \

...minated in the first round; $B$ would then defeat $C$ in the second round. Note, however, that $A$ defeats $B$ by $6$ votes to $3$, and also defeats $C$ by

(note that this sum is finite). The result is a field, denoted by $ k(( X)) $.

one may note, for the sake of comparison, that the de Moivre–Laplace theorem yields 64

Given an infinite bi-ideal sequence, we note that each $f_i$ is a prefix of $f_{i+1}$ and so the $f_i$ converge to an in

...><td valign="top">[a7]</td> <td valign="top"> S. Saks, "Addition to the note on some functionals" ''Trans. Amer. Math. Soc.'' , '''35''' (1933) pp. 9

...iterion, see [[#References|[2]]], [[#References|[3]]]). It is important to note that if the equations of some of the elements of the systems are unknown, t

(Note the way of controlling the sequencing of rules application by means of the

...TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> P.L. Walker, "A note on an inequality with non-conjugate parameters" ''Proc. Edinburgh Math. So

Note that $ \sigma _ {d} ( A) \subset \sigma _ {r} ( A) $,

...quasi-Frobenius (cf. also [[Quasi-Frobenius ring|Quasi-Frobenius ring]]). Note that, if $\operatorname{ad} _ { q } \in L$, then $\operatorname{ad} _ { q }

..., $z^*=x-iy$ are introduced instead of the coordinates $x,y$ in the plane; note that the Laplace potential operator (cf. [[Laplace operator|Laplace operato

Note that if $n=m$ the formula reduces to

Note that the maximum-likelihood estimator is an M-estimator, obtained by puttin ...} ( - b , x ) )$. This $\psi_b$ has come to be known as Huber's function. Note that when $b \downarrow 0$, this M-estimator tends to the median (cf. also

...are mappings preserving $\otimes$, arbitrary $\vee$, and $\top$; and also note that $\mathbf{FRM}$ and $\textbf{SFRM}$ embed into <img align="absmiddle" b

...with $w=abcba$. One can produce the following sequence of transformations. Note that the starting tableau has the "reading property" mentioned above.

...</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D.W. Babbage, "A note on the quadrics through a canonical curve" ''J. London. Math. Soc.'' , '''1

...s $p$-semi-simple if $X = X_p$, and $X$ is a BCK-algebra if $X_p = \{0\}$. Note that for a $p$-semi-simple BCI-algebra, the closed ideals are precisely the

...pivotal role in the ordinary triangle inequality, by a triangle function. Note that for a function $ d $ ...ant probabilistic pseudo-metric space is a transformation generated space. Note that $ F _ {pq } = F _ {f ( p ) f ( q ) } $,

Note that it is the upper semi-circle and can also be given by (the upper part o

...obtained by solving a non-linear equation. All other problems are linear. (Note: the theory may also be built around $ R _ {l} $,

...This was already stated in the seminal work of I.M. Gel'fand around 1940. Note that there is a deep related question. Let $A$ be a [[Fréchet algebra|Fré ...s. Indeed, it is known just which Banach algebras arise in this situation. Note that the proof of this theorem requires the assumption of the [[Continuum h

...a rational number but is an integer when $F$ is a Frobenius endomorphism. Note that more generally we say a finite group is a ''finite group of Lie type'' ...e a semidirect product decomposition $\mbf{P} = \mbf{U} \rtimes \mbf{L}$, (note in particular we have $\mbf{L}$ is contained in the [[Normalizer of a subse

...aluate, as it depends on three arguments: $\alpha$, $\beta$ and $\theta$. (Note that $\theta$ is not restricted to be a Euclidean parameter; it can also be ...fficiency of $T$ with respect to $V$, with notation $e ^ { s } ( T , V )$. Note that

when $\theta$ varies in $\Theta ( \mu )$. Note that $\mu ^ { \prime } \in \mathcal{M} ( E )$ is such that the two sets $F ...replaced by $\bar{X} _ { n } = 1 / n ( X _ { 1 } + \ldots + X _ { n } )$. Note that since $M _ { F }$ is an open set, and from the strong [[Law of large n

...space-times (cf. [[Penrose cosmic censorship|Penrose cosmic censorship]]). Note that in the physics literature there are various conflicting definitions of

...></tr><tr><td valign="top">[a8]</td> <td valign="top"> R. Fleissner, "A note on Baire 1 Darboux functions" ''Real Anal. Exch.'' , '''3''' (1977-78) {

Note that every completely-continuous mapping (or [[Compact mapping|compact mapp

...imply $x \ast z \in I$. It is known that always $(x \ast z) \ast z \in I$. Note that in a positive implicative BCK-algebra, every ideal is implicative. Imp

Note that the adjoint of the X-ray transform can be traced back to the Buffon ne

...istic and applies only in the case where $X$ is smooth. It is important to note that this is the only known method in mathematics to prove the existence of ...ram $X \rightarrow Y \leftarrow X ^ { + }$ is called a flip (or log flip). Note that $- ( K _ { X } + B )$ is $\phi$-ample.

.../td></tr><tr><td valign="top">[a2]</td> <td valign="top"> K. Borsuk, "A note on the theory of shape of compacta" ''Fund. Math.'' , '''67''' (1970) pp

Meyer's note particular, we note his "Essai sur une exposition nouvelle de

Note that $ L _ {0} $)

...old|Complex manifold]]) not admitting any [[Kähler metric|Kähler metric]] (note that, for $ n = 1 $,

Note that $ n! = w ^ {n} = 1 $

For a more comprehensive historical note on $\mathcal{A}$-sets (also called analytic sets) see Rogers' contribution

...permits efficient sparse matrix factorizations. Having said that, however, note that Broyden's method can be effective on certain very large problems [[#Re To start with, note that it can be assumed that $B _ { 0 } = I$. The reason for this is that if

...D></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E.A. Michael, "A note on paracompact spaces" ''Proc. Amer. Math. Soc.'' , '''4''' (1953) pp. 8

...called Zipf's law was first observed by J.B. Estoup [[#References|[a2]]]. (Note that Zipf's role was not so much the observation itself, which, by the way, ...imately at the same time, R.H. MacArthur [[#References|[a10]]] published a note on a model for the relative abundancy of species. His model of "non-overla

...discovery of his famous axioms of order (see [[Pasch axiom|Pasch axiom]]). Note that already C.F. Gauss [[#References|[a9]]] noted that the word "between" ...\emptyset$ together with a mapping $d : S \times S \rightarrow \mathbf R$. Note that for a pre-Hilbert space, $S = X$ and $d ( x , y ) = \| x - y \|$.

...7</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> B. Chen, "A note on LPT scheduling" ''Oper. Res. Lett.'' , '''14''' (1993) pp. 139–142<

where $\phi$ is an isometry with $\phi ^ { 2 } = \operatorname{id}$. Note that in dimension $m = 4$, there is an algebraic curvature tensor which is

...{ i }$ and $\theta _ { i }$, in order to approximate an unknown function. Note that for each $\theta \in \mathbf{R}$ and $\mathbf{w} \in \mathbf{R} ^ { n

Note that $ \xi ^ {i} ( p)= ( X x ^ {i} ) ( p) $;

Note that the two last conditions just say that $x$ is feasible and that the thi

setting of this note. Standard textbooks on Markov chains are {{Cite|K,MT,N,R}}. $ \mathbb{E} [ \tau_{ i , i } ] < \infty $ for at least one $ i \in S$. Note that for $p=1/2$ the random walk is recurrent and for $ p < 1/2 $ it

...metric upper [[Semi-continuous function|semi-continuous function]] on $P$. Note that, conversely, the Zahorski property does not imply the Luzin–Menshov

There are several important features of this spectral sequence. First, note that $ E _ {s,t } ^ {1} $

...h excludes the existence of any effective criteria in these terms. One may note that in all the cases mentioned here there exist finite complete systems, a

Note that the families $ \mathfrak B _ {x} $

It is important to note that the next to the last equation above comes from integrating by parts an Note that $e _ { i }$ is the multi-index of all zeros except for a $1$ in the $i

...of Banach spaces depends on the geometry of the space. It is important to note that this is a two-way connection, in that the probabilistic properties oft

such spaces are sometimes called Priestley spaces. Note that a Priestley space $ \mathop{\rm spec} A $

...\times X\to X$ and $G\times Y\to Y$. Each action defines the partition<ref>Note that any two orbits $G(x)$ and $G(x')$ in $X$ are either disjoint or coinci

...ity of an|Algorithm, computational complexity of an]]). It is important to note that both the computational complexity of the algorithm and the complexity ...valign="top">[22]</TD> <TD valign="top"> A. Church, "Correction to "a note on the Entscheidungsproblem" " ''J. Symbolic Logic'' , '''1''' : 3 (1936

...ften called ''Baire-1 functions'' and arise naturally in several problems (note, for instance, that the derivative of a differentiable function is a Baire-

...rded by Katz [[#References|[Ka]]], justifies (a belief in) the conjecture. Note that hypersurfaces of any given degree <math>m</math> determine a projectiv ...math>(2,2,2,1,1)</math> determine a one-parameter family of nodal curves. (Note that the incidence type is misprinted in [[#References|[Vo3]]].)

..."La trivialité topologique n'implique pas les conditions de Whitney" ''Note C.R. Acad. Sci. Paris Ser. A'' , '''280''' (1975) pp. 365</td></tr><tr><t

...ry as a whole, but on one of its parts only. M.V. Keldysh was the first to note the dependence of the formulation of a boundary value problem on the nature

...'if and only if any one of the following six equivalent conditions holds. (Note:'' $\mx{V}$ ''may be replaced by its Moore--Penrose inverse'' === Note ===

note that it yields the arithmetic mean when there are a set of direct |valign="top"|{{Ref|2}}||valign="top"| Legendre, A.M. (1820). Note par M.*** Second supplement to the third edition of Legendre (1805), pp.79

Note that any functional-differential equation is equivalent to a hybrid system ...given interval $[ - h ( t ) , - g ( t ) ]$ or $( - \infty , - g ( t ) ]$. Note that $x _ { t }$ may be treated as the fragment of the function $x$ at the

...e [[#References|[a9]]] and [[Cohomology operation|Cohomology operation]]). Note that for $p=2$ this is simply the squaring operation.

Note that all above-mentioned functors in algebraic geometry are contravariant.

==== '''Note:''' ====

...62</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> C. Jacobi, "Note sur l'intégration des équations différentielles de la dynamique" ''C.R.

|valign="top"|{{Ref|AM}}|| S. Albeverio, Z.M. Ma, "A note on quasicontinuous kernels representing quasilinear positive maps" ''Forum

...TR><TR><TD valign="top">[10]</TD> <TD valign="top"> M. Nagata, T. Miyata, "Note on semi-reductive groups" ''J. Math. Kyoto Univ.'' , '''3''' (1964) pp. 379

...) = \operatorname { sup } \{ t \geq 0 : g_t ( u ) \text { is defined} \}$. Note that $l ( u ) = \infty$ means that $g _ { t } ( u )$ is defined for all $t

Note that $\kappa = 2 J$ if $f ( 0 ) \neq 0$, and $\kappa = 2 J + 1$ if $f ( 0 ) Note that the data $\mathcal{S}$ allow one to find a unique $f ( k )$ by solving

<TD valign="top"> J. Isbell, "A note on complete closure algebras" ''Math. Systems Theory'' , '''3''' (1969)

Plackett and Barnard note, "interspersed with friendly advice on both Another aspect of this paper worthy of note is Gosset's use of a

...cdots \wedge \mathrm{d}{x^{n}} $ defined by $ g $. For any function $ f $, note that $ f \omega $ is an $ n $-form, so $ \displaystyle \int_{M} f \omega $

...$ with the one-dimensional Lebesgue measure. This example is trivial, but note the different σ-fields: neither $\F_{\om'}\subset\F$ nor $\F\subset\F_{\ ...ly finite (not necessarily probability) measures {{Cite|P}}, {{Cite|B}}. Note also "standard σ-finite measure" in {{Cite|Mac}}. Some authors exclude s

Note that if $ m > n $,

...t $d$ is a vector and the function $p(x)$ is $\Gamma$-periodic. One should note that in physics the vector $k=-id$ is called the quasi-momentum [[#Referenc

...- f )$ if the link is given by an M/M/1 queue (cf. also [[Queue|Queue]]). Note that $t$ decreases if $k_b$ increases, for $f$ fixed. Suppose some flow dem

...nt in $\Omega$. Here, all $q_j$ and $s _ { j }$ are non-negative integers. Note that the integral moments $c _ { q }$ depend on $f$ and $\Gamma$, but the m

...y simultaneous access to a single memory location by different processors. Note however that simultaneous access to different memory locations in unit time

Note that, by the reduction, the power product $xy^2$ occurring in $p$ is replac

...e theorems in a $B$-space setting. Before briefly examining some of these, note that there exist subnormal operators $A$ and $B$ for which $\operatorname {

...</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> R. Ellis, "A note on the continuity of the inverse" ''Proc. Amer. Math. Soc.'' , '''8''' (1

...calls $P$ the centre on $C$ of the members of $\mathfrak { R } ( C , P )$; note that $R ^ { \prime } ( P ) = R ( P )$ $\Leftrightarrow$ $P$ is a simple poi

Note that D. Sarason [[#References|[a26]]] gave an interesting proof of the Juli ...td></tr><tr><td valign="top">[a27]</td> <td valign="top"> J. Serrin, "A note on harmonic functions defined in a half-plane" ''Duke Math. J.'' , '''23''

Note that b) is an immediate consequence of a).

..., if $A\not\mapsto M$ and $B\not\mapsto M$ imply $A\times B\not\mapsto M$. Note that a graph $A$ has a good $n$-colouring if and only if $A\mapsto K_n$. He

...></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J.M. Borwein, "A note on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or

of [[#References|[a1]]]. Note that an arbitrary lift will lead to multiplying the period by a power of $

...nt problems in the theory of linear operators on indefinite metric spaces. Note that, generally speaking, such semi-groups are not everywhere differentiabl Note also that in the one-dimensional case it follows from the [[Maximum princip

Note that this implies that the fixed-point set $X ^ { P }$ has finitely many co

One calls the function $h$ the '''hamiltonian'''. Note that for a symplectic manifold, viewed as a Poisson manifold, this definiti

It is important to note that the instability is controlled by the wave-number $ k $

by a Dehn surgery. Note that $ M _ {1} $

...</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> S. Newcomb, "Note on the frequency of use of different digits in natural numbers" ''Amer. J.

...rm{PG}(3,q)$ is a set of $q^2+1$ points, no three of which are collinear. (Note that in [[#References|[a2]]] and some other older publications, an ovoid is

Note that

|valign="top"|{{Ref|AM}}|| S. Albeverio, Zh.M. Ma, "A note on quasicontinuous kernels representing quasilinear positive maps" ''Forum

...$ denote the set of natural numbers starting at $1$ (i.e., excluding $0$). Note that neither of the special cases below does exclude the simple generic cas

note now when women's roles are more clearly appreciated, that the history of It is interesting to note that Chebyshev's eminence was late in being

...hm} must be suitably interpreted, since $v$ is not necessarily continuous. Note that the almost everywhere differentiability of $v$ is not sufficient to gu

This note is a brief description of that theory. and a short note such as this cannot even begin to do

commute. (Note that a natural numbers object is just a list object over $1$.) Given these,

Note that if iii) (or, equivalently, iv)) were to hold for all $h$, not just tho

algebra generated by the random variables between the braces. Note that $ \beta $-

...( T _ { j } - ) = 1 \}$ with $T _ { j }$ now the $j$th observed failure. (Note that in contrast to the situation with non-random covariates described abov

...there is a $ C^{r} $-section $ f: V \to X $ such that $ \phi = J^{r} f $; note that $ \phi $ determines $ f $ uniquely (if it exists). The fine topology o

...X$, $Y$ are normed vector spaces, the above problem is not easy to answer (note that, in this case, taking $r = 1$ is no loss of generality). For example,

...></tr><tr><td valign="top">[a7]</td> <td valign="top"> C.L. Mallows, "A note on asymptotic joint normality" ''Ann. Math. Stat.'' , '''43''' (1972) pp

...sposed matrix of $H$ and $ { I } _ { n }$ is the unit matrix of order $n$. Note that the problem of constructing Hadamard matrices of all orders $n \equiv

...$ is called a Bose–Mesner algebra (briefly BM-algebra) if $A _ { 1 } = I$. Note that according to E. Bannai and T. Ito [[#References|[a1]]], a homogeneous

<tr><td valign="top">[a2]</td> <td valign="top"> C. Fefferman, "A note on spherical summation multipliers" ''Israel J. Math.'' , '''15''' (1973)

.../TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R. Narasimhan, "A note on Stein spaces and their normalisations" ''Ann. Scuola Norm. Sup. Pisa'' ,

...coincides with the approximate derivative, whereas the opposite is false (note, for instance, that if $g$ coincides almost everywhere with $f$, then $g$ i

...</td></tr><tr><td valign="top">[a6]</td> <td valign="top"> P. Turán, "A note of welcome" ''J. Graph Th.'' , '''1''' (1977) pp. 7–9</td></tr><tr><td

...td></tr><tr><td valign="top">[a18]</td> <td valign="top"> T. Kashiwada, "A note on a Riemannian space with Sasakian 3-structure" ''Nat. Sci. Rep. Ochanomiz

...[a1]</TD> <TD valign="top"> W. Auzinger, R. Frank, G. Kirlinger, "A note on convergence concepts for stiff problems" ''Computing'' , '''44''' (199

...\infty } X$. In fact, one way to construct the Brown–Gitler spectra is to note that the functor

Note that $\mathcal{P} \subset \mathcal{NP}$; the question whether equality hold

...the system returns to the state of stable equilibrium. It is important to note in this connection that the system passes from stable equilibrium to auto-o

|valign="top"|{{Ref|CE}}|| Copeland, A. H.; Erdős, P. "Note on normal numbers", ''Bulletin of the American Mathematical Society'' '''52

Note that $ A _ {q} ^ \infty ( D,G ) $

We note that the monotonicity property ...call it a Lindenbaum matrix. (The full definition will be given later.) We note that an operator $Cn$ satisfying $(c_1)-(c_3)$ can be obtained from a "clos

...n. Of the most important results obtained in axiomatic set theory one must note Gödel's result on the consistency of the [[Continuum hypothesis|continuum One should note that these two axiom systems, $ \Sigma $

...tudied analytic structure in fibres in $\text{I} _ { p }$-spaces. However, note that there is a peak set (cf. also [[Algebra of functions|Algebra of functi

...D valign="top">[a2]</TD> <TD valign="top"> J. Karhumäki, I. Simon, "A note on elementary homomorphisms and the regularity of equality sets" ''EATCS B

<TR><TD valign="top">[a4]</TD> <TD valign="top"> J. Isbell, "A note on complete closure algebras" ''Math. Systems Theory'' , '''3''' (1969)

...p"| Gani, J. (1982). The early use of stochastic methods: an historical note on McKendrick's pioneering papers. In ''Statistics and Probability: Essa ...gn="top"| Whittle, P. (1955). The outcome of a stochastic epidemic - a note on Bailey's paper. '' Biometrika'' '''42''', 116-122.

...></tr><tr><td valign="top">[a3]</td> <td valign="top"> R.R. Bahadur, "A note on quantiles in large samples" ''Ann. Math. Stat.'' , '''37''' (1966) pp

Note that when $H ( \theta , X ) = \theta - X$, and $\gamma _ { n } = 1 / n$, $\

...></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> H.J. Bremermann, "Note on plurisubharmonic and Hartogs functions" ''Proc. Amer. Math. Soc.'' , ''

...on to the theory of surreal numbers" , London Mathematical Society Lecture Note Series '''110''' Cambridge University Press (1986) {{ZBL|0595.12017}}</TD>

..., the notions of "m-accretive" and "maximal accretive" are equivalent. Note that in Hilbert spaces, "accretive" is also known as "monotone" .

...exist and $\psi$ is the appropriate section for this $\Spin_\C$ structure. Note that $A$ is just a $U(1)$ Abelian connection, and so $F = dA$, with $F^+$ b ...] for the generating function which, for $M$ of simple type, reads (though note that the bundle denoted by $L$ here corresponds to the square of the bundle

...= \pi ( n )$, where $\pi ( m )$ denotes the set of prime divisors of $m$. Note that a simple well-known argument shows that a finitely-generated residuall (Note that there is a minor error in the actual statement in the original [[#Refe

...rithms from [[Linear-algebra(2)|linear algebra]] are especially important. Note that numerical algorithms for standard infinite-dimensional linear problems

...></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> R. Schwabauer, "A note on commutative semi-groups" ''Proc. Amer. Math. Soc.'' , '''20''' (1969)

Note that if $ \mathfrak G $

...P )$, and it will be called the local number of generators of $M$ at $P$. Note that the right-hand side of (a1) is equal to the [[Dimension|dimension]] of

Note that $\pi_\L$ are precisely all irreducible finite-dimensional representati ...form|Killing form]] on $\fg$ normalized by the condition $(H_0|H_0) = 2$. (Note that for $\fg = \def\fsl{\mathfrak{sl}}\fsl_n(\C)$, one has $(x|y) = \tr xy

...ism theorem, which provides a statistical version of structural stability. Note that the billiard system is very sensitive to initial conditions and the pe

Note that every <img align="absmiddle" border="0" src="https://www.encyclopediao Note that if <img align="absmiddle" border="0" src="https://www.encyclopediaofma

Note that the central problem of [[Spectral synthesis|spectral synthesis]], that

...( u ) =$ least non-negative integer not among the values $g ( F ( u ) )$. Note that $g ( u ) = 0$ if $u$ is a leaf of $G$, i.e., if $F ( u ) = \emptyset$.

consists of one boundary element. Note that the Green function of an open Riemann surface $ R $,

...ect to the property $\operatorname { Ext } _ { H } ^ { 1 } ( T , T ) = 0$. Note further, that a tilting module $T$ over a hereditary algebra is uniquely de

...t{const}$. Denote $L_m(z):=\sum^{J}_{|j|=0}a_{jm}z^j$, $z\in\textbf{C}^n$. Note that $L_m(e^{zx})=e^{zx}L_m(z)$, $z.x:=\sum^n_{j=1}z_jx_j$.

...t [[#References|[a9]]] and the Matusita coefficient [[#References|[a10]]]. Note that $ \rho ( {\mathsf P} _ {1} , {\mathsf P} _ {2} ) $

...</TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> F. Faà di Bruno, "Note sur une nouvelle formule du calcul différentiel" ''Quart. J. Math.'' , ''

Note that there is a subspace $\mathfrak{h}_R$ of $\mathfrak h $ that is, in the

equivalently, the category of (left) $ A $-modules, is an Abelian category. Note that if $ e $ provided there exists an irreducible mapping $ X \rightarrow Y $ (note that a mapping $ f: X \rightarrow Y $

...If $\overline{a},\overline{b}\in A$, choose representatives $a,b\in H$ and note that $(\overline{a},\overline{b})=\chi(aba^{-1}b^{-1})$ is independent of t

..." referred to above are called frames or locales (see [[Locale|Locale]]). Note that the binary operation $ \supset $

...lgebraic stability plays an important role in the theory of B-convergence. Note that algebraic stability implies B-stability. For non-confluent methods, i.

It is essential to note that the quantity $K$ defined in (a1) may "be equal to" $+ \infty$.

have a TAR model driven by (or excited by) $\{Y_t\}$. Note that the X_{t-i}^2, \;\; \alpha_0 >0, \alpha_i \ge 0, i = 1, \ldots, p.$ Note

...\in \mathbf{F} [ t ]$ such that $p ( T ) = 0$ (cf. [[#References|[a2]]]). Note that I. Kaplansky in [[#References|[a1]]] considered rings with a polynomia

...singular holomorphic foliation on $\C P^2$ with a non-identical holonomy. Note that according to this definition, the same leaf may carry many different l

...w w_1,\ldots, x_n\leftarrow w_n]$ can be defined analogously. One has to note, that the substitutions of $x_1,\ldots,x_n$ with $w_1,\ldots,w_n$ are exec

note that $ \cap S $

...ied with the Taylor polynomials of order $k$ of objects from $\mathscr M$. Note that these jet approximations are affine finite-dimensional spaces (the Tal

Now, note that the wave operators $\Omega _ { \pm }$ which map $\psi _ { \pm }$ to $\

It is interesting to note that when $ M $

Note, however, that in much of the literature the AIC is defined as minus the ab Note that although the BIC is based on an entirely different approach it differs

Note that the probability distribution $ \mu _ {t} ^ {x} $

...</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top"> N. Wiener, "Note on paper of O. Perron" ''J. Math. Phys.'' , '''4''' (1925) pp. 21–32

...d></tr><tr><td valign="top">[a9]</td> <td valign="top"> R. Nakagawa, "A note on $( E , M )$-functors" H. Herrlich (ed.) G. Preuss (ed.) , ''Categorica

L. Fejér [[#References|[a2]]] was the first to note the importance of the class of trigonometric polynomials that assume only n

...stic subgroup was introduced in 1958 by H. Wielandt [[#References|[a18]]]. Note that $\omega ( G ) = G$ if and only if $G$ is a [[T-group]], i.e. normality

...ilow's theorem). There is no such representation in dimensions $n \geq 3$. Note that for all dimensions $n \geq 2$ and $K &gt; 1$ the set of those points w

Note that here the completed tensor product is used.

note that the latter is of the form $ \mathbf R \times D $,

...ates partial differentiation with respect to $\dot { x } \square ^ { r }$. Note that the Einstein [[summation convention]] on repeated upper and lower indi

Note that $A$ being self-adjoint implies conservation of energy. So the problem

...s $x _ { 1 } , \ldots , x _ { n }$ to $g_{k}$ is the depth of the circuit. Note that formulas are circuits whose graphs are trees.

...ch uses a homotopy double groupoid of a pair of spaces. It is important to note that group objects internal to the category of groups are just Abelian grou

...ution|empirical cdf]] associated with the sample $ (X_{1},\ldots,X_{n}) $. Note that $ \hat{F}_{n} $ (which is a random step function) puts a probability m

...which are Abelian categories (cf. [[Abelian category|Abelian category]]). Note that the second of these classes of varieties is closed under Morita equiva

...has a counterpart for many-to-one maps. (See {{Cite|S|Sect. 4.5}}.) First, note that an arbitrary map $f:X\to Y$ is a composition of the projection $p:X\to

are the same. Note that the semi-simple Artinian rings are characterized by the fact that they ...gular local rings that play an important part in basic algebraic geometry. Note that a local commutative ring is regular exactly then when it has finite gl

|valign="top"|{{Ref|5}}||valign="top"| Newcomb, S. (1881). Note on the frequency of use of different digits in natural numbers. ''American

...es. This was described in [[#References|[a12]]] for $k = 0$ (see Table 1). Note that one of the measures must be the Jacobi or the Laguerre weight function

...TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> K.L. Chung, "A note on the ergodic theorem of information theory" ''Ann. Math. Stat.'' , '''32

...fficiently one has to understand how the process $Q(t)$ behaves over time. Note that all $A(t), D(t)$, and $Q(t)$ are stochastic processes (which are seque ...rate. The ratio of arrival rate to service rate is the traffic intensity. Note that we have assumed the processes to be time-independent.

one obtains the usual residue. Note that $ \psi dz $

...bedded in a topological nearness space (i.e. symmetric topological space). Note that the category $ \mathop{\rm Top} _ {s} $ is not topological (note that the imbedding of the Hausdorff space $ \mathbf Q $

Note that the integrand above is the density function associated with $ n $

historical note, and he had some from the

...in trying to prove the Bieberbach conjecture, the following are worthy of note.

...ijdam (2010), followed by a final response Morgan et al. (2010). This last note incidentally finally revealed Craig Whitaker's original wording of the prob Note the use of the trick of symmetrization -- randomization over the door initi

...\dots , f _ { n } \in \mathcal{D}$. When combining (a1) and (a2), one can note that this limit-measure $\mu$ then accounts for the time-ordered $n$-point

Note that redundant relations cannot be added, since by Tietze's theorem (see [[ ...$K ^ { 2 } \times I \searrow \operatorname{pt}$, where $I$ is an interval. Note that Z) also implies AC), as $K ^ { 2 } \nearrow K ^ { 2 }\times I \searrow

Note that the canonical parametrization by $\theta$ generally differs from a mor

...package for the computation of a few eigenvalues of a large sparse matrix. Note that realistic applications of such matrices, i.e. in stability and bifurca

$h^2$. Note that, sometimes, extrapolation algorithms are able to sum

It is interesting to note that the predicate $ P ( \mathfrak A , f ( x _ {1} \dots x _ {n} )) $(

extended to products by the quasi-triangularity axioms. Note that $R$ need not obey the Yang–Baxter equations in order to have a bi-al

its probability may be estimated as $(m + 1)/(m+n+2)$. Note that

...localized version of the product of a distribution and a smooth function. Note that $( \delta ( x ) , \text { vp } 1 / x ) \notin \mathcal M _ { 1 } ( \ma

...over free monoids, as shown in [[#References|[a6]]]. It is interesting to note that only the third solution extends to a multi-dimensional variant of the

...discrepancy tractable", Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 284, 211-266 (2001). {{ZBL|0978.65014}}</td></tr>

Note that the lemma deals with function spaces of many variables. For functions

...R><TR><TD valign="top">[a13]</TD> <TD valign="top"> V.Yu. Shavrukov, "A note on the diagonalizable algebras of PA and ZF" ''Ann. Pure Appl. Logic'' , '

to much derision. In a confidential note to the Rector at

...ous functions in two variables applied to the coefficients $x,y$ and $z$? (Note that every function in a finite number of variables is a composite of disco

...{ 2 } \tau ( G ) = \operatorname { det } ( J + L )$. It is interesting to note that this determinantal way of computing $\tau ( G )$ requires $\nu ^ { 3 }

which is a much simpler task. It is important to note that in this

It is interesting to note that he cites only the monograph

Note that the [[Axiom of choice|axiom of choice]] is an assertion about the exis

is exact. Note that the objects of $ {\mathcal C} _ {S} $

...me. For $0 \le i \le d$, put $n_i = p_{ii}^0$, and $n = \sum_i n_i = |X|$. Note that for each $i \in \{1,\ldots,d\}$ for which $R_i^\top = R_i$, $(X,R_i)$

(Note that the left-hand side of (a2) is the so-called distribution function of

.../TD></TR><TR><TD valign="top">[a35]</TD> <TD valign="top"> G.S. Rinehart, "Note on the global dimension of a certain ring" ''Proc. Amer. Math. Soc.'' , '''

It is, finally, interesting to note that some algebras arising in genetics are Jordan algebras [[#References|[1

...n, the joint selection probabilities ${\pi }_{ij}$ appear in the variance. Note that when estimating the parameters of a finite population using the design ...tition the population into$ a=A\cap B^c$, $b=A^c\cap B$, and $ab=A\cap B$. Note that the population total Y may be expressed as $Y=Y_a+Y_b+Y_{ab}$, where $

First note that the search direction $d ^ { k }$ is the negative gradient of $f$ with

Note that the link and cumulant functions for each of the above GLM log-likeliho ratios for the logistic model. Note the fact that 1st class passengers had a near 6 times greater

...ator]] and $\tilde { a } ( e ^ { i \theta } ) = a ( e ^ { - i \theta } )$. Note that for sufficiently smooth and invertible matrix functions $a$ the operat

...mpletely in [[#References|[a4]]], see also [[#References|[a3]]], Sect. 36. Note also that if $F : \mathcal{M} f \rightarrow \mathcal{M} f$ is a bundle func

.../td></tr><tr><td valign="top">[a72]</td> <td valign="top"> P. Turan, "A note on the real zeros of Dirichlet L-functions" ''Acta Arith.'' , '''V''' (19

...|valign="top"| Dietz, K. (1988). The first epidemic model: a historical note on P.D. En'ko. ''Australian Journal of Statistics'' '''30A''', 56-65.

...these auxiliary problems converge to the solution of the original problem. Note that in an unconditional minimization problem every direction is feasible,

Gertrude Cox of my staff." This terse note was to have far-reaching

For a historical perspective, note that the basic idea of equal-degree factorization was known to A.M. Legendr

Note that the normalized exponential $ \Phi ( \omega ) = N \mathop{\rm exp} ...]] with a large test function space and, consequently, less distributions. Note also the paper of P. Krée in [[#References|[a5]]] for an overview and refe

This biographical note is adapted from an abridged version of

with a fixed polynomial $p \in P ( k )$. Note that $\{ p _ M\}$ defined by (a7) satisfies (a6). The Eichler cohomology gr

Note that this identity immediately implies

...defined as for the two-player conflict in terms of evolutionary stability. Note that the definition, while only referring to two strategies, implicitly dea

To conclude, note that it is possible to define Dirichlet eigenvalues for much more general d

Note that there is another field in which collapsibility is prominent, and which

...algebra. (See [[#References|[a2]]] for an expanded survey.) Of particular note is the notion of the Connes spectrum of an automorphism group [[#References

cf. [[Descriptive set theory|Descriptive set theory]]) suffices. Note that, by a theorem of Tarski, the complete theory of the standard model of

Note that for each $i$, $\operatorname { span } \{ e _ { i } , f _ { i } , h _ {

...topological spaces the definitions of {{Cite|D}} and {{Cite|H}} disagree. Note also the σ-algebra of [[universally measurable]] sets.

...ropped from $\lambda _ { p }$ and $b _ { p } ^ { ( 2 ) }$ due to symmetry. Note that

...geometric interpretations of the classical Ramanujan–Petersson identities. Note that for Selberg's conjecture one again assumes that $\pi_\infty$ is of cla

Note that the choice $\Phi = ( N ! ) ^ { - 1 / 2 } \operatorname { det } f _ { j

...cant research areas can be listed in this connection. It is interesting to note that these areas also reflect the historical development of computer scienc

...} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Milnor, "A note on curvature and fundamental group" ''J. Diff. Geom.'' , '''2''' (1968) pp.

Note that an earlier definition of rotation of a set-valued field $\Phi x = x -

...method|saddle point method]] or the residue theorem. It is interesting to note that the asymptotic expressions thus obtained may be used as guides, "model

...x proves that the set $M$ of all sets does not exist. It is interesting to note in this connection that there exist axiomatic systems of set theory, such a

...on these restrictions on $m$ and $n$ are assumed, unless otherwise stated; note that this estimate $n \geq 2 ^ { 48 }$ was improved on by I.G. Lysenok [[#R

...ive root for $E$, and the polynomial $f$ is called a primitive polynomial. Note that these terms carry a different meaning in the context of Galois fields

To describe $V ^ { * }$ in the complex topology, first note that the Harish–Chandra realization [[#References|[a6]]] of $X$ as a boun

Note that in a rigid braided category, morphisms <img align="absmiddle" border="

...)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> T. Onoyama, "Note on random distributions" ''Mem. Fac. Sci. Kyushu Univ. Ser. A'' , '''13'''

may note today that D'Alembert had pinpointed some of the very real

...Caccioppoli sets, we will not assume that the set $E$ has finite measure: note that anyway the quantity $V({\bf 1}_E, \Omega)$ is well defined. |valign="top"|{{Ref|DL}}|| C. De Lellis, "A note on Alberti's rank-one theorem", Transport equations and multi-D hyperboli

Note that a Horn clause like $ ( A, \neg B _ {1} \dots \neg B _ {k} ) $

...as $ E = E' \oplus 1 $, where $ 1 $ is the one-dimensional trivial bundle. Note that $ E' $ is usually not uniquely defined. Moreover, if the rank of $ E $

It is important to note that the vectors $\bf H$, $\mathbf{B}$ and $\bf M$ are axial, while the vec

...nly if its dual unit ball in the weak-${}^{*}$ topology is Gul'ko compact. Note that a Corson compact space, or Corson compactum, is, by definition, a comp

Note that $ \mu ( a ) = 0 $

Note also that the Zhukovskii–Kutta–Chaplygin condition is usually called th

|valign="top"|{{Ref|Ku}}||valign="top"| W. Kulpa, "A note on the dimension Dind" ''Colloq. Math.'', '''25''' (1972) pp. 227–240

papers, it is interesting to note an attempt at small

where $f \in A ( D )$ and $g _ { u } \in A / B$. Note that (a7) gives a more concrete description of the duality than does (a5).

...he first approach to a mathematical solution of the given problem, one can note the independent arguments of functions; these are usually the three coordin

....g., Hölder and Zygmund classes) as well as certain locally convex spaces. Note, however, that there are continuous functions $\phi$ for which $\mathcal{A}

called failure functions (note that $ \phi _ {1} = \phi $

...tion systems, i.e. systems to the base two, three, etc. It is important to note that none of the constructions of the theory of real numbers given above (a

...ns to higher dimensions as the braided geometry of quantum-braided planes. Note that non-trivial co-products of the above linear form on the generators are

...s drawn from a finite population. ''Biometrika'', '''17''': 472--479. (The note on this republication reads ``These results with others were originally pub

Note that the various names given to various variants of these problems are by n

...ft lokaler Ringe" , ''Lecture Notes Math.'' , '''634''' , Springer (1978) (Note: The proof of (3.1.1) is wrong) {{MR|0485851}} {{ZBL|0401.13013}} </td></tr

...en with the same properties, [[#References|[a11]]], [[#References|[a14]]]. Note that if $e$ could be taken as a projection, e.g. the range projection of $e

[[K-theory|$K$-theory]]. It is important to note in this connection that a projective module can be regarded as the module o

...g process, a turning point test and a fundamental inequality: A historical note on I.J. Bienaymé. ''Biometrika'' '''59''', 680-683.

Note that since $\sigma _{\operatorname{T}}$ is defined in terms of the actions

Soviets. Its last number bears the note: \, publisher: A. Chuprov. In his obituary for

Note that $ T _ {n-1} ( k) [ z _ {n} ] = k \langle z _ {1}, \dots, z _ {n-1}

Note that $ \mathop{\rm Cosk} ^ {0} p = B $.

...nd explicitly features the [[Eigen value|eigenvalues]] on the diagonal. Note that this form holds only over an algebraically closed field $\Bbbk$, e.g Note that the three empty sets $\varnothing_i$, are different from the algebra

for $\mathbb{P}_n$ the law of $X^{\epsilon_n}$.) Note that $I(\phi)=0$ if and only if $\phi$ is a solution of the

...d} R$ of finite-dimensional right $R$-modules (see [[#References|[a17]]]). Note that $q_R = q_Q$ if $R = K Q$.

...proximity weight, proximal connectedness, etc. G. Cantor was the first to note the simple property of proximal connectedness, viz. $ A \times ( P \setmi

Note that $R$ satisfies (a1) if and only if $B = \tau _ { V , V } R$ satisfies

...n]] $\Gamma\subset TE$ complementary (transversal) to the ''vertical''<ref>Note that the vertical distribution is [[integrability|integrable]] ([[involutiv

note that $ \nabla \cdot {\vec{h} } = 0 $

very difficult. It is interesting to note that in the course of a

Note that in the above decomposition it is not assumed that the <img align="absm

...6</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> B.L. Welch, "Note on discriminant functions" ''Biometrika'' , '''31''' : 1–2 (1939) pp.

See also [[#References|[a4]]] for an introduction to root systems. Note that the notion of Coxeter graph used in the article above differs slightly

Note the important fact that the plane $ \mathbf R ^ {2} $

Note that $k^{\down r} = 0$ for positive integers $r > k$. Note that the matrix $\theta$ and its transpose determine the same

...></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> O.B. Widlund, "A note on unconditionally stable linear multistep methods" ''Nordisk. Tidskr. Inf

Note that when $g = 0$, the equation reduces to the dispersionless Korteweg–de ..., \Lambda = 1 ) = a _ { i } ( \hat { u } _ { k } , \Lambda = T _ { 1 } )$ (note $T _ { 1 } \sim \Lambda$ in the Seiberg–Witten situation). Then one defin

...lly, $\mathcal{H}$ specifies $\zeta _ { 1 } = \ldots = \zeta _ { q } = 0$. Note that $\zeta _ { q + 1} , \dots , \zeta _ { r }$ are unrestricted throughou ...y their confidence intervals simultaneously with probability $1 - \alpha$. Note that (a7) is stated without needing the canonical system so that the confid

::Note that the term "analysis" in that article is the traditional term for the th

Note that $L_{\overline{0}}$ is a [[Lie algebra|Lie algebra]] in the standard se

...) = \alpha ( z ) \alpha ^ { * } ( w ) - \beta ( z ) \beta ^ { * } ( w )$. Note, in particular, that for $R = I$ one gets $R ( z , w ) = 1 / ( 1 - z w ^ {

behind a given set of observations. The final note of the

obscurity of his style. However in preparing this note I found that

Note that the error is dominated by $ {\widehat{k} } ^ {2} ( A ) $,

with $v$ as the infinitesimal generator<ref>Note that all iterates (resp., flow maps) are defined only as ''germs'', thus th Note that this normal form is integrable in quadratures.

odds ratios. Note that the link function is the log of the odds of

...</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> Yu.D. Burago, "Note on the isoperimetric inequality on two-dimensional surfaces" ''Siberian Mat

It is important to note that there exists an upper bound for exact slopes

|valign="top"|{{Ref|Bl}}||valign="top"| S. Bloch, "A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer con

...e expansion of the Weyl symbol $a$. In the same vein, it is also useful to note that when considering pseudo-differential operators acting on half-densitie

This note appears on page iv of the pamphlet Playfair

Note that in the algebraic category one requires that the points "at infinity" a

To explicitly construct $\mu$, note that since $\operatorname { rank } M = \operatorname { rank } M ( n ) = r$

Note that if $ G = U ( 1 ) $,

...able periodic solutions (i.e. stable limit cycles), [[#References|[a15]]]. Note that $p$ is fixed in a model, unlike $\gamma$, which is a free parameter. R

Indeed, ignoring questions of convergence for the moment, note that (by the unique factorization into prime elements of $G$) the series

...is not at a complete statistical equilibrium. Secondly, it is important to note that the gas is situated in its own gravitation field, which should be dete

A.N. Korkin and E.I. Zolotarev [[#References|[14]]] were the first to note the isolation phenomenon in this case (which was also the first case of the

...1}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> A. Cayley, "Note on the theory of permutations" ''Phil. Mag. (3)'' , '''34''' (1849) pp. 527

...</td></tr><tr><td valign="top">[a35]</td> <td valign="top"> Y. Shimoda, "A note on Rees algebras of two-dimensional local domains" ''J. Math. Kyoto Univ.''

of $\def\P{\mathbb{P}}\P^{r-1}(C)$. Note that $\O^2 = \{(\omega_1/\omega_2)\} = C-K_\infty$, which is the Drinfel'd

...ave to maximize the latter equation for $\boldsymbol{\theta}$ numerically. Note that this maximization problem is a lot simpler than the maximization of th

...unit (so-called associated numbers) have one and the same ideal factors. (Note that ideal numbers are defined relative to the original field $ k $ ; for

...actly of the members of members of $x$. This union is denoted by $\cup x$. Note that for two sets $a$ and $b$, $\cup \{ a , b \}$ is the usual union $a \cu

...03}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> I.D. Ado, "Note on the representation of finite continuous groups by means of linear substi

Note that any Möbius transform, or stereographic projection, or tubular $ \ep

First of all, one must note that the non-applicability of solving by the trivial method is only one of

<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W.H. Flemming, "A note on differential games of prescribed duration" , ''Contributions to the theo

...etail [[#References|[a15]]], [[#References|[a13]]], [[#References|[a17]]]. Note that as suggested by the presence of chaotic solutions and by a Painlevé a

...er the algorithm; here, $\theta ^ { * }$ is the limit of the EM sequence — note that $Q ( \theta | \theta ^ { * } )$ and $\int f ( \theta , \phi ) d \phi$

Note that both for the investigations by Hilbert, and for much later investigati

...lizations of the [[Algebra of logic|algebra of logic]]. It is important to note that in the algebra of logic, propositions take only two truth values ( "ye

...nd Yau on the theory of three-dimensional manifolds, [[#References|[21]]]. Note that if one has a theorem on the existence of a minimal solution in the hom

Note that any <img align="absmiddle" border="0" src="https://www.encyclopediaofm

Here it's important to note that, as he stated it, Cauchy's condition for continuity, alike with Bolzan ...lled "an academic dogma."<ref>Blaszczyk, Katz, & Sherry, p. 23 The authors note challenges to this "dogma" ranging from S. Feferman’s predicative concept

Note that the short-run effect of advertising here is $d S _ { t } / d u _ { t }

science; but of particular note is the remarkable effort of L. J. Savage

''Note''. A subbundle of a trivial bundle may well be nontrivial.

One may finally note that the logically unobjectionable definition of the concepts connected wit

One should note that the first derivative term is multiplied by a factor that may become ze

...ould be weaker). Both hardware and software implementations are available. Note that Merkle and Hellman's so-called knapsack cryptosystem [[#References|[a1

Finally, a note on some quasi-mathematical matters that are purposely not discussed in this (In ordinary language, the forms E and O have the alternate forms shown. Note that by introducing an additional Copula Verb '''are not''' and using the e

...D}$-filters of $\mathbf{A}$ to the lattice of congruences of $\mathbf{A}$. Note that the Leibniz and Suszko congruences coincide on $\mathcal{D}$-filters j