User talk:Ramon Roca

(indice: 3D protein interactions)

Ramon Roca

Dept. of Macromolecules (CNB-CSIC) &amp; Dept. of Biochemistry and Molecular Biology (UB).

A SET THEORY OF INFORMATION: MODES OF VIBRATION (biblio) (bioinformatics : protein interactions : molecular computation)

integers (mass/energy) vs. primes (information/entropy) Molecular Computing: Prime numbers (G&ouml;del &amp; Golay/Best), Mendeleev &amp; Riemann.

 R. Landauer;  Computation A fundamental physical view. ''Phys. Scr.''35, 88-95 (1987).

  There really is no software, in the strict sense of disembodied information, but only inactive and relatively static hardware. Thus, the handling of information is inevitably tied to the physical universe. Evolution and the origin of life can be viewed as an optimization process, in which fluctuations (e.g., mutations) take us from one metastable ecology, to a new one. We might, with equal justice, refer to the revolution of an electron around a hydrogen nucleus, or the rotation of a water wheel, as self organization.  Dyson : Extraordinary! Do you realize that's the pair-correlation function for the eigenvalues of a random Hermitian matrix? It's also a model of the energy levels in a heavy nucleus [erbium-166 (protons+neutrons)].

(Isotopes_chart.pdf) [In Heisenberg's formulation of quantum mechanics, the internal state of an atom or a nucleus is represented by a Hermitian matrix whose eigenvalues are the energy levels of the spectrum.]

<td style="vertical-align: top; font-family: georgia; width: 10%;"> "Performing good theory implies to attend to contexts, that are always particular. And these require interdisciplinarity, the use of induction and some epistemological pretensions more modest than those supposed by abstract theories of illustrated root. Criticism from Frankfurt School was referred to social character questions. [...] In front of the greek intellectualism, that admitted persuassion as daughter of good reasoning, Gorgias proclaim that its efficacy was rather within the words employed in reasoning. Humans, said Benjamin, tend to "overname" things by means of abstractions and generalizations."

Ferran Requejo (ferran.requejo@upf.edu), LaVanguardia 23/VI/2007 NUMBER THEORETICS AND INFORMATION LEVELS

 Levin and Chaitin definition of algorithmic entropy : A program is self-delimiting it if needs no special symbol, other than the digits 0 and 1 of which it is composed, to marks its end; discrete objects other than binary strings, e.g., integers, or coarse-grained cells in a continous phase space, may be defined by indexing them by binary strings in a standard way (as Monte Carlo program describes distributions of macrostates).  Fundamental Theorem of Arithmetic : Every natural number is uniquely decomposable into a product of prime powers. Primes are the building blocks (factors) of the positive integers: The prime integers of an integer determine its properties.  Euler's function  [cos &Icirc;&cedil; + j sen &Icirc;&cedil; = ej&Icirc;&cedil;] describes vibrations (waves) connecting geometry (trigonometric functions -the integer values n, in real space-) and algebra (exponential function -the prime factors p, in reciprocal space-) -consider: n + n' = p -.

Any arbitrary shape can be regarded as a locus of intersecting surfaces of nth order (generated in terms of Fourier descriptors). The total amount of information embodied in simple shapes is determined by the shape category, the number of surfaces and the form or order of the surfaces-defining equations (e.g., quadratic, cubic, etc) (Ayers, 1994)</a> - see table</a> -: Badii utilized the idea that chaos is made up of combinations of the periodic orbits. A 'primitive' is defined by two conditions: it is periodic (i.e., in the infinity sequence there are arbitrarily long repetitions of it) and it cannot be broken down to other primitives. The [hierarchical] tree [structure] is constituted by the primitives and their admissible combinations (the first level, the primitive combination; 2nd level, pairwise combination; n-th level, n-ary combinations, and so on). (Kampis, 1991)</a>. With G&ouml;del numbering (by instance, using the original symbols as exponents of a prime factorization) it becomes possible to encode and decode state transformations to and from states directly. This procedure, which is itself algorithmic, ensures the existence of a dynamics without the use of any further information. All we need is a component-system which produces them by complexity-increasing procedures. (Kampis, 1991)</a>.

Golay codes are based upon prime numbers: Golay codes G23 (binary, [23,12,7]2, 3-correcting) and G11 (ternary, [11,6,5]3, 2-correcting) are perfect, systematic and linear. Theorem of Best: All perfects codes over any alphabet, with p&acirc;&#8240;&yen;3 and p&acirc;&#8240; 6,8 are equivalent to Rep2(n) or G23. (Brunat, 2001)</a> This theorem implies two basic levels of information (2 and 23). Similarly, in music, it is possible to observe these same levels: first one, ternary, it would be formed by values 0, 1 and 2 and they correspond to silence, semitone and tone; second level depends on "musical temperament", so, considering temperated scale, it is formed by 12 notes. Another example results from comparing binary (2 elements, first level) and decimal (10 elements, second level) systems; in this case, adding a third level (20, 100, 1000), although related to logarithmic scales, it is usually done on an arbitrary and subjective manner. <td style="vertical-align: top;"> Number of elements per period (Mendeleiev table):

<td style="vertical-align: top;">

<td style="vertical-align: top;"> Types of electronic orbitals or (sub)shells: <td style="vertical-align: top;"> Reinterpreting theorem of Best, i.e., if G23 is a prototypical perfect code, it could be stablished the next periods for prime numbers: <small style="color: rgb(255, 0, 0);"> +++ (because of  "technical" reasons, it is considered 1 as the "first" prime, i.e., n=1) First period (elements 1 and 2) defines the even-odd concepts, equivalent in signal transmition, to renormalized [-1, 0, 1], or in musical terms to semitone and tone. Second period (8 elements, primes from 3 to 23: increment=20) defines -consolidates- the primity concept and is equivalent, for instance, to 2 musical scales, or to binary Golay code G23. Third period (8 elements, primes from 29 to 59: increment=30) determines the self-organizing character of the progression of the prime numbers: 29 is (almost 30) a  "prime multiple"  of  3.

With only 18 elements, it is incremented the "informative value" from a signal type 2 to another signal type 59 (almost 60). In music, considering equal-tempered scales, it is possible to divide the octave into 59 intervals to approximate the frequency ratios from just intonation; so, it is also a good choice the standard division of the octave in 12 intervals, as 5 octaves of 12 notes (i.e., 60 notes is quite near of 59: just a semitone) permit to "construct a quasiperfect code" (as extended Golay codes G24 and G12 can be generators of perfect codes G23 and G11). From an "evolutionary" point of view (as increment of information complexity), it must be considered the three first elements as critical (and concentrical): 1 defines unity and appears as "opposition" to 0 (algebraically, a "trivial" vectorial subspace); 2 defines pair/even, as first distinctive of symmetry (1+1);

3 defines "primity" concept in its orthodox sense, considered as another distinctive of (a/self)symmetry (2+1), conditioning asymmetrical interactions. The most important detail is just the increase of complexity: as the number of elements grows, "the growing itself becomes faster". Essentially, this increasing is due to main basic relationships of the three first elements (2+1, 3+2), that, altogether with the fourth element (5, just a consequence of the previous relationships), conform the first "extended period". Another approach is to consider "3+2" interaction as the combination of ternary and binary systems, a basic subset of complexity in terms of information.

Analogous to Golay code G11, parameters [11,6] (respectively, dimension and length) could be extended to other concrete/discrete pairs [p,n], being p prime and n its integer (the number of protons); the third parameter of Golay codes, related to error correction, is named "Hamming minimum distance" [could be equivalent to the number of neutrons?]. Then, Carbon is represented as a "ternary code" with parameters C [11,6]; first fourth elements are defined as the next pairs (vectors, matrices, codes): Hydrogen, H [1,1]; Helium, He [2,2]; Lithium, Li [3,3]; and Beryllium, Be [5,4].<br style="font-weight: bold;"> <ul> </ul> From these premises: a.- It is represented, by FFT phases of generated inverse sine waves, "p orbitals" ( p spheres of information ) of the first 118 elements and others, so some elements, like Er (atomic number: n=68; p= 331), Rn (86,439) and predicted Ac' (121,659), "reorganize their orbitals into N beams".

b.- Simple arithmetical rules are used for different molecules (from water to ATP; resp., p=19 and p=541). My initial question was if hydrogen and ATP could be compared "quantitatively"  at entropy and information levels; within "primity scale", H and ATP information values differe in a "logarithmic order 1:101". My proposal implies that different levels of information are organized in a complex system within the basis of prime numbers, as a result from considering as "Golay codes" the atomic elements, and then, distributed by "Medeleiev periods ". COMPLEX SYSTEMS: MOLECULAR COMPUTATION

In biology, different molecules interact by means of their composing elements and determine diverse informative values by their respective surfaces and volumes. For example, H+ and Ca2+ are two types of signals at the subcellular level; at protein level, 20 aminoacids are classified depending on their characteristics (analysable through Markov models or equivalence matrices). So, it seems logical to consider each atom as a characteristic signal type, quantizable, normalized by "G&ouml;del numbers". Evolutionary, increment of complexity appears because of the periodical progression (Mendeleiev) of prime numbers, and because of interactions that appear due to increasing possibilities and elements. Prime integers are hierarchical strange attractors (characteristic or typical tones) of complex self-organized (harmonized) systems. <img alt="" src="tesis/log.jpg" style="height: 277px; width: 711px;" title="">

[BINAIRES VS. TERTIARIES 0,1 &lt;&lt;==&gt;&gt;-1,0,1 (0,1,2): complexity and self-organized systems] Then, systems can be classified as simples; ([N-1, N] or [N , N+1]) or complexes [N-1, N , N+1]. Dynamics appears when the strange attractor implies [P1 , N, P2 if N=P1+1 and N=P2-1 (so, P1+2 = P2 and P2-2= P) ;

being N an integer number, and P, a prime one.</b> <hr style="width: 100%; height: 2px;"> Results and other conjectures Atom prime chart:</a> <img alt="" src="tesis/atomchart.jpg" title="atom prime chart" style="border: 3px solid ; width: 700px; height: 285px;"></a> Digital signal processing Phase analysis (Lissajous Plot graphs ***</a>) of prime generated tones:

<ul style="margin-left: 40px;"> <li> Inverse sine (pure tone: fundamental, no harmonics); </li> <li> Audio signals in Hz. </li> </ul> Non prime: Control data and Dynamics</a> Prime phases of atomic elements (p orbitals): [original size]</a> &amp; [reduced view]</a>

"Nobles": "Erbium-like":

More prime phases: part I (661 to 1427)</a>; part II (1429 to 7919)</a>; part III (7927 to 21059)</a>. </a> Dynamics, transitions and polarization

"Erbium-like" series (local strong attractors: the Riemann series) <h3 style="font-weight: bold;"> <i>Uncoupled and coupled terms are related to even and odd (1/2 or N/N+1), determining the L (levo) or R (dextro) character {anti-matter/matter, to be or not?}.</i> <hr style="width: 100%; height: 2px;"> Volume is a function of distribution of mass-energy quanta Integers: Atomic numbers (electrons and protons) represent mass/energy quanta. Primes: Every prime is associated to its integer, and represents a standarization of atomic volume/surface relationship, defining atomic information quanta.

<a href="tesis/atomos_graph.htm">Atomic typology (prime vs. integer):</a> <a src="tesis/atomos_graph.htm"><img src="tesis/atomos_graph.htm"></a> <a href="tesis/atomos_graph.jpg"><img alt="" src="tesis/atomos_graph0.jpg" style="border: 0px solid ; width: 1032px; height: 758px;"></a> <span style="text-decoration: underline; color: rgb(51, 204, 0);">Prime series of atomic elements

<img alt="" src="tesis/group_numbers.jpg" style="border: 2px solid ; width: 917px; height: 586px;" title="">  Periods K to Z20 [n/p (log p - log n)] 

<td colspan="1" rowspan="1" style="font-weight: bold; text-align: center;"> <td colspan="5" rowspan="1" style="vertical-align: top; width: 60%;"> <img alt="" src="tesis/zzlog.jpg" style="width: 550px; height: 348px;"> a pictorial view of Theorem of Best for Golay codes <td colspan="2" rowspan="1" style="vertical-align: top; width: 20px; white-space: nowrap; font-weight: bold; text-align: right;"> e= &#931; (1+1/n)1/n

[e**=&#931;(1+1/p)1/p] e ~ 2.72 &#960; ~ 3.14      <span style="color: rgb(51, 204, 0);">2/6&#960; ~ 1.645

&#960;-e ~ 0.423 1-2/3 ~ 0.333 1-e/&#960; ~ 0.1347 e&#960;i = -1 i2 = -1

[==&gt; &#960;= 2(ln i)/i ](?) "Prime interactions"

<th style="text-align: center;">

<td style="text-align: center;">

<td style="text-align: center;">

<td style="vertical-align: top;"><img style="width: 720px; height: 540px;" alt="" src="tesis/molecules.jpg">

Atoms vs. molecules <td style="vertical-align: top; text-align: center;"> <img alt="" src="tesis/molecules.jpg" style="width: 720px; height: 540px;"> <hr style="width: 100%; height: 2px;"> <hr style="width: 100%; height: 2px;">

NOTES <a name="Lissajous_figure"></a>Lissajous figure [from <a href="http://wims.unice.fr/">wims</a>] Lissajous curve is a parametric curve in dimension 2, described by a pair of functions x = cos(nt), y = sin(mt+a). [from "Collins English Dictionary", 1984]

A curve traced out by a point that undergoes two simple harmonic motions in mutually perpendicular directions. The shape of these curves is characteristic of the relative phase and frequencies of the motion; they are used to determine the frequencies and phases of alternating voltages [C19: named after A. Lissajous (1822-1880), French physicist]. <a href="tesis/coding_theory.html">CODING THEORY </a> <a href="tesis/G11.jpg">Ternary Golay code G11.</a> <a href="tesis/Golay.jpg">Extended ternary Golay code G12.</a> <a href="tesis/theorem_and_proofs.html">"Perfect e-error correcting code" Theorem (</a><a href="file:///D:/the_prime_project/tesis/theorem_and_proofs.html">Tiet&auml;v&auml;inen and Van Lint, 1971) </a>

<a href="tesis/theorem_and_proofs.html"></a><a href="tesis/theorem_and_proofs.html">and proof tools (sphere packing condition &amp; Lloyd's theorem).</a> <h2 style="font-weight: normal;"> <a href="tesis/coding_history.html">From "Ten milestones in the history of source coding theory":</a> <b> Discovery of the Lloyd algorithm. </b> An N-level vector quantizer for k-dimensional blocks of real-valued source samples is designed by determining the N

codevectors in k-dimensional Euclidean space into which the source blocks are to be quantized. The goal in vector quantizer design is to find a vector quantizer for which the expected squared-error quantization noise achieves a value close to the minimum. Stuart Lloyd, in an unpublished technical report written in 1957 (eventually published in 1982 [14]) proposed an algorithm for accomplishing this goal. (Although Lloyd stated his algorithm for the scalar quantization case in which k = 1, it trivially extends to the k &gt; 1 case.) The Lloyd algorithm starts with an initial quantizer and modifies it through a sequence of iterations--on each iteration, the codevectors from the preceding iteration are replaced with the centroids of the nearest neighbor regions corresponding to these codevectors. As long as successive iterations of the Lloyd algorithm continue to generate new quantizers, the quantization noise strictly decreases. (This is because of the Lloyd-Max necessary conditions for an optimal quantizer, which appear to have been first discovered not by Lloyd or Max but by the Polish mathematicians Lukaszewicz and Steinhaus [15].) In the scalar quantization case, if the density function of the source samples is log-concave and has finite variance, it is known that the N-level quantizers generated by iterates of the Lloyd algorithm have asymptotically optimal performance, independently of which N-level quantizer is chosen at the start of the iteration process. Unfortunately, the Lloyd algorithm is sensitive to the choice of initial quantizer if k &gt; 1: for some choices, the quantization noise may not decrease to the minimum value. Recent research (cited in [11]) has focused on modifications of the Lloyd algorithm through simulated annealing or stochastic relaxation techniques that avoid this problem. It should be mentioned that there have been other significant contributions to the area of quantization in addition to the Lloyd algorithm--there have been numerous advances in high rate quantization theory and in lattice quantization, for example. The reader is referred to the recent text [7] for a thorough discussion of quantizer theory and design.

''' Discovery of Lempel-Ziv codes. ''' The well-known universal noiseless source coding technique due to Jacob Ziv and Abraham Lempel was announced in 1977 [26], although the method is based upon a notion of string complexity that had been proposed by these two authors in a paper the year before. With probability one, a stationary ergodic finite-alphabet source generates a sequence which, when encoded using the Lempel-Ziv algorithm, yields a compression rate equal to the entropy rate, asymptotically as the number of source samples goes to infinity. The Lempel-Ziv algorithm is the most important noiseless source coding technique in the entire history of source coding. A spate of papers has been devoted to the theoretical and practical aspects of Lempel-Ziv coding. On the theoretical side, perhaps the most significant of these is the recent paper by Ornstein and Weiss [18]. <a href="tesis/biblio.html#LZW">Digital Signal Processing: LWZ Compression</a> <a href="tesis/Riemann_zeta_function.html">The spectrum of Riemannium</a> Brian Hayes (<a href="http://www.americanscientist.org/">American Scientist</a> July-August, 2003; vol. 91 (4), 296:300 )

<a href="tesis/number_theory_and_physics.html#pball">Prime numbers not so random? (Phillip Ball, 2003).</a> <a href="tesis/number_theory_and_physics.html#mwatkins">Surprising connections between number theory and physics (M. Watkins, 2004).</a> <font face="Arial,Verdana,Helvetica,Geneva,Sans-Serif">"<i>Are the prime numbers in a self-organized critical state?</i>" <p style="margin-left: 160px;"> <font face="Arial,Verdana,Helvetica,Geneva,Sans-Serif">M. Wolf, "1/f

noise in the distribution of prime numbers", Physica A 241 (1997), 493-499. <font face="Arial,Verdana,Helvetica,Geneva,Sans-Serif">P. Bak, C. Tang, and K. Wiesenfeld, "Self-organized criticality", Physical Review A 38 (1988), 364-374.

Data (audible spectrum: 16.4 - 21096 Hz; ~ 10 scales) <a href="tesis/scale_frequencies.html">FREQUENCIES AND WAVELENGTHS FOR EQUAL-TEMPERED SCALE</a>

 Upupa epops  (hoopoe, abubilla, puput), bird song:

Definitions: Cardinal Numbers are positive integers (counting numbers) that represent "how many?" Ordinal Numbers are numbers that describe position: first, second, third, fourth,... last.

(Cardinal numbers -real space-:) Set of the atomic elements and their atomic/electronic -integer- numbers (and, by extension, molecules): {H1, He2, ..., C6, N7, O8, ..., Na11, Mg12, ..., K19, Ca20, ..., Fe26, ..., Zn30,...}. (Ordinal numbers -reciprocal space-:) G&iuml;&iquest;&frac12;delization, in prime numbers (nuclear types -harmonic fundamental-): {H1, He2, ..., C11, N13, O17, ..., Na29, Mg31, ..., K61, Ca67, ...,Fe97, ...,Zn109,...} See <a href="tesis/atomos_graph.htm">atomic typology graph</a> and <a href="tesis/atomos_list.html" src="tesis/atomos_list.html">table</a>: Representation in log<font size="-1">10 is explicit (it also should result interesting log<font size="-1">2 [log<font size="-1">3

, log<font size="-1">5 , log<font size="-1">7 ], ln or log<font style="font-family: symbol;" size="-1">p ).

Waves &amp; Atoms: Melody or tone: Succession (atom). Harmony or accord: Simultaneous combination (molecule). Rythm: Relative regular groups (Mendeleiev: a code of infinite period, or, better, an infinite series of finite sets/rings-). Musical language considers an octave as 7 notes (12 semitones), while harmony is constructed classically (major scale or ionian mode) from 3rd  and 5th  chords (the 2nd

is equal to a whole note -step-); jazz added 11th and 13th, i.e., some additional complexity appears as considering 2 octaves -24 semitones-, because within a 7-based code, 11 and 13 are equal to 4th and 6th steps. Modes result from different step-patterns or sequences of the notes (modality, its own sound characteristics).   Then, relationships between notes (symbols) conform different phrases and motives (codes), and their interactions (grammar), including environmental noise, can result onto characteristic (sounds, music, words, silence) languages. It is not coincidence the true relationship between music and prime numbers: waves and their interactions need some "harmony" (proportions between attractor types or prime levels).    Atoms and molecules can be described as concrete sets of electrons and their respective nuclei, i.e., as integer numbers (atomic or electronic numbers at Mendeleiev table). Moreover, because of  "volume is a function of distribution of mass-energy quanta", let's consider each atom as a "perfect code of a characteristic type" -a prime G&iuml;&iquest;&frac12;del? number- (due to specific volume/surface relationships, n interacting electrons conform different wave configurations or information/entropy classes). Here, it is intented to describe interactions as "fract[ion]al relationships" of different nuclear and molecular types. So, each atom or molecule is represented as a binary vector (n protons, p-order -information/entropy level-); for each electronic (atomic or molecular) wave equation, there is a typical wave [prime] value and their "aritmethic operations" (physico-chemical interactions or level transitions) are due through mass/energy quanta (as Bethe cycle is a "set of integer operations"). Stability and dynamics are then referred to a "concrete set of prime levels" (the components of the complex system), as progressive "harmonic chords" (or augmented, disminished, sustained, ...).

Computability, constructibility, self-reproduction and evolvability are concepts easily applied to biomolecular systems. Complexity-increasing procedures (physics, chemistry, biology) and "privileged zero" { l } states (quark, foton, electron, energy, temperature, noise) have digitized interactions, although at different levels. Cell metabolism transferres the energy from 1 proton (hydrogen) onto a molecule with 260 protons (ATP), both quantic wave states with different information/entropy architectures ("magnitude orders" or level codes). Proteins or nucleic acids also have "quantizable" descriptions and interactions (H+ transferences, ...). As biopolymers (DNA, proteins) can be stabilized by helix conformations, it is clear that this characteristic property (self organization) is related to the electronic wave nature of atoms and molecules. The genetic code permits to connect thermodynamical, morphological and control information (as well as symbolic), by combining DNA sequences (4 nucleotides) and proteins (20 aminoacids) plus environment (water, ions, other molecules and physicochemical conditions) as a whole (cell metabolism). Each organism is defined as a set of genetic information; each protein is a 3D structure of a concrete sequence of aminoacids; each molecule is a peculiar set of atoms; each atom is a characteristic string of subatomic particles. <img alt="" src="tesis/log1.jpg" style="height: 540px; width: 720px;"> Piano schema could symbolize as the prime series evolves, beyond G23 subset, as an alternance of groups formed by 2 and 3 sets of elements (or, more precisely, as  a growing function of  3rd and  4th levels, as showed by Medeleiev periodicity - 3x(2+2p) - and Pythagorean temperament - 9/8 -).

(hard) Hypotheses: <span style="font-family: times new roman,times,serif;"> G23 and G11 are usually constructed from their respective relatives G24 and G12 (implicitly, G23 includes minor sets as G19, G17, G13, G11, G7, G5, G3; while G12 permits to contruct G11 and G13). Theorem of Best shows that there exist different orders of symmetry distributed by periods: even-odd (1-2), prime (3-23); or: ln2, ln 10, ln 18, ln 36, ln 54, ln 86, ln 118, etc (maybe: log2

, log10, log18 , log36 , log54 , log86, log118 , etc). Best's theorem implies 2 main levels of complexity: the set {2,1,0} and G23 (a basic set of primes), as any other set  referred to any upper value can be described as even-odd, n-tuple or prime (the third level of complexity, only considering the whole infinite series).

As music is a "cyclic code" ({1,2,3,5,7} ~ 11/13 [or 17/19/23]) -compare with Golay codes-, it could be considered: <ul> <li> stars (H + He), binary base [pseudo-ternary]: codes {[0*,] 1, 2}. </li> <li> life, base H / C / N / O / P / S: codes {1 / 11 / 13 / 17 / 43 / 47}. </li> <li> na&iuml;&iquest;&frac12;fly, three (basic sets of) "universes" (matter classes), in hierarchical evolution (in fact,  z  "universes", interacting anisotropic or enharmonically -fractal proportions between different orbitals, dimensions or degrees of freedom): <ul>

<li> elementary particles: l {0} -"trivial" vectorial subspace-, </li> <li> plasma: K {1 / 2} and

</li> <li> the rest: L,M,N,O,P,Q,... {3 / 5 / 7 / 11 / 13 / 17 / 19 / 23 /... /  z }. </li> </ul> </li> <li> Quarks are related to "subharmonic" values as 1/2, 1/3, 1/5, 1/7, 1/11, 1/13 (or to another basic set of primes as G23). </li>

<li> Radioactive isotopes (see <a href="http://chemlab.pc.maricopa.edu/periodic/isotopes.html">http://chemlab.pc.maricopa.edu/periodic/isotopes.html</a>) are "no tonic" atoms. </li> <li> Unstable molecules are "no harmonic" (with the environment). </li> </ul> <hr style="width: 100%; height: 2px;"> <hr style="width: 100%; height: 2px;"> <hr style="width: 100%; height: 2px;"> <hr style="width: 100%; height: 2px;">