The document discusses numeric partitions and graphs involving "Eugênio Numbers" (EN), "Krishna Numbers" (KN), and functions related to the number e and other constants. It defines EN and KN sets and describes partitioning them into "generations" based on digit counts. Equations are given for EN functions involving operations like addition, multiplication and logarithms. Graphs are proposed plotting the ratios of "Aleph Red" to "Aleph Omega" for different values of variables like ∆.
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...IJMER
In this article, 𝜆 − 𝑗𝑒𝑐𝑡𝑖𝑜𝑛 has been introduced which is a generalization of trijection
operator as introduced in P.Chandra’s Ph. D. thesis titled “Investigation into the theory of operators
and linear spaces” (Patna University,1977). We obtain relation between ranges and null spaces of two
given 𝜆 − 𝑗𝑒𝑐𝑡𝑖𝑜𝑛𝑠 under suitable conditions.
Dual Spaces of Generalized Cesaro Sequence Space and Related Matrix Mappinginventionjournals
In this paper we define the generalized Cesaro sequence spaces 푐푒푠(푝, 푞, 푠). We prove the space 푐푒푠(푝, 푞, 푠) is a complete paranorm space. In section-2 we determine its Kothe-Toeplitz dual. In section-3 we establish necessary and sufficient conditions for a matrix A to map 푐푒푠 푝, 푞, 푠 to 푙∞ and 푐푒푠(푝, 푞, 푠) to c, where 푙∞ is the space of all bounded sequences and c is the space of all convergent sequences. We also get some known and unknown results as remarks.
Slides from the talk on recurrent networks and LSTMs at SV AI and Big Data Association meetup. A full video of the talk—https://www.youtube.com/watch?v=TiHpdp4QC6k.
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...IJMER
In this article, 𝜆 − 𝑗𝑒𝑐𝑡𝑖𝑜𝑛 has been introduced which is a generalization of trijection
operator as introduced in P.Chandra’s Ph. D. thesis titled “Investigation into the theory of operators
and linear spaces” (Patna University,1977). We obtain relation between ranges and null spaces of two
given 𝜆 − 𝑗𝑒𝑐𝑡𝑖𝑜𝑛𝑠 under suitable conditions.
Dual Spaces of Generalized Cesaro Sequence Space and Related Matrix Mappinginventionjournals
In this paper we define the generalized Cesaro sequence spaces 푐푒푠(푝, 푞, 푠). We prove the space 푐푒푠(푝, 푞, 푠) is a complete paranorm space. In section-2 we determine its Kothe-Toeplitz dual. In section-3 we establish necessary and sufficient conditions for a matrix A to map 푐푒푠 푝, 푞, 푠 to 푙∞ and 푐푒푠(푝, 푞, 푠) to c, where 푙∞ is the space of all bounded sequences and c is the space of all convergent sequences. We also get some known and unknown results as remarks.
Slides from the talk on recurrent networks and LSTMs at SV AI and Big Data Association meetup. A full video of the talk—https://www.youtube.com/watch?v=TiHpdp4QC6k.
To find the complete solution to the second order PDE
(i.e) To find the Complementary Function & Particular Integral for a second order (Higher Order) PDE
Periodic Function, Dirichlet's Condition, Fourier series, Even & Odd functions, Euler's Formula for Fourier Coefficients, Change of Interval, Fourier series in the intervals (0,2l), (-l,l) , (-pi, pi), (0, 2pi), Half Range Cosine & Sine series Root mean square, Complex Form of Fourier series, Parseval's Identity
The aim of this paper is to study the existence and approximation of periodic solutions for non-linear systems of integral equations, by using the numerical-analytic method which were introduced by Samoilenko[ 10, 11]. The study of such nonlinear integral equations is more general and leads us to improve and extend the results of Butris [2].
In this paper, the concepts of sequences and series of complement normalized fuzzy numbers are introduced in terms of 𝛾-level, so that some properties and characterizations are presented, and some convergence theorems are proved
To find the complete solution to the second order PDE
(i.e) To find the Complementary Function & Particular Integral for a second order (Higher Order) PDE
Periodic Function, Dirichlet's Condition, Fourier series, Even & Odd functions, Euler's Formula for Fourier Coefficients, Change of Interval, Fourier series in the intervals (0,2l), (-l,l) , (-pi, pi), (0, 2pi), Half Range Cosine & Sine series Root mean square, Complex Form of Fourier series, Parseval's Identity
The aim of this paper is to study the existence and approximation of periodic solutions for non-linear systems of integral equations, by using the numerical-analytic method which were introduced by Samoilenko[ 10, 11]. The study of such nonlinear integral equations is more general and leads us to improve and extend the results of Butris [2].
In this paper, the concepts of sequences and series of complement normalized fuzzy numbers are introduced in terms of 𝛾-level, so that some properties and characterizations are presented, and some convergence theorems are proved
Differential Geometry for Machine LearningSEMINARGROOT
References:
Differential Geometry of Curves and Surfaces, Manfredo P. Do Carmo (2016)
Differential Geometry by Claudio Arezzo
Youtube: https://youtu.be/tKnBj7B2PSg
What is a Manifold?
Youtube: https://youtu.be/CEXSSz0gZI4
Shape analysis (MIT spring 2019) by Justin Solomon
Youtube: https://youtu.be/GEljqHZb30c
Tensor Calculus
Youtube: https://youtu.be/kGXr1SF3WmA
Manifolds: A Gentle Introduction,
Hyperbolic Geometry and Poincaré Embeddings by Brian Keng
Link: http://bjlkeng.github.io/posts/manifolds/,
http://bjlkeng.github.io/posts/hyperbolic-geometry-and-poincare-embeddings/
Statistical Learning models for Manifold-Valued measurements with application to computer vision and neuroimaging by Hyunwoo J.Kim
Sources:
Visual - various maths sites (credits to original creator)
Questions - Dong Zong's Textbook
suitable for SUEC (Maths), SPM (Maths and Add Maths) too
Theory of Relativity
Maybe travelling in time is an interesting topic. Also the idea of the flow of time at high speeds is a difficult idea to understand. But did you know that in 1905, someone dared to think differently. He is Albert Einstein. Questions such as, what will you see if you are moving at the speed of light? Well, it is argued that light speed is the maximum speed that is available in the entire universe. The speed of light was calculated by Maxwell using the equations of Electromagnetic wave.
c=√(1/(ε_o μ_o ))
We were able to understand that anything that has speed travels a certain distance in space in amount of time.
Einstein argued that measurements done on physically observable quantity must be uniform in all inertial reference frame. The problem is there is no such as universal reference frame. This gives rise to the assumption that everyone is moving relative to one another. This would give rise to another claim that is, “measurements taken from one reference frame, will be different from measurements taken from other frame of reference”. This argument is absurd because it will mean that laws of physics were different for different reference frames. The theory of relativity holds to the fact that the laws of physics were the same for all inertial reference frames.
This will be eminent when we apply the concept of the Doppler Effect to sound. We know that whenever the source of the sound moves with a velocity V_s, with respect to the observer there will be a change in the measured frequency. Furthermore there will be more measurements that can be made depending on the observer. So how do we determine the real frequency of the sound emitted by the source?
Another instance is when we are on board a plane with some velocity Vplane and we fire a bullet the relative velocity of the bullet on an stationary observer will be;
V=Vbullet+Vplane
Which is correct in Galilean transformation. Now what if we turn on the headlight of a plane? Would it mean that the speed of light will be the velocity of the plane + the speed of light? (v=c) ?. Absolutely not, because this will violate the premise that, “the speed of light is constant in a vacuum”.
Clearly from the two instances there must be a different formula that will unify measurements made on different reference frame. This method is called transformation.
So let us create two equation that will unify measurements in these two instances. The first instance is at the plane, the observer at the plane will have (x,y,z,t). and the observer from the earth will us the coordinates (x^',y^',z^',t^'). So which is it the spaceship is moving away from the earth or the earth is moving away from the spaceship. To fix this, we assume that the origin O and O^'coincide and are parallel to one another at all times. Further more we let t and t^' be equal that is t= t^'.
and more....
Recurrence relation of Bessel's and Legendre's functionPartho Ghosh
This presentation tells about use recurrence relation in finding the solution of ordinary differential equations, with special emphasis on Bessel's and Legendre's Function.
Utilitas Mathematica Journal original research and review articles. Utilitas Mathematica Journal commits to strengthening our professional community by making it more just, equitable, diverse, and inclusive. Algebra ,Analysis ,Geometry Offers selected original research in Pure and Applied Mathematics and Statistics.
Left and Right Folds - Comparison of a mathematical definition and a programm...Philip Schwarz
We compare typical definitions of the left and right fold functions, with their mathematical definitions in Sergei Winitzki’s upcoming book: The Science of Functional Programming.
Errata:
Slide 13: "The way 𝑓𝑜𝑙𝑑𝑙 does it is by associating to the right" - should, of course ,end in "to the left".
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
7. Princípio de Trans100dência
Todo Algébrico Irracional é raiz de um polinômio não nulo, com coeficientes inteiros. Tal polinômio
tem um comprimento dado pelo somatório do valor absoluto de cada um desses seus coeficientes:
𝑳(𝑷) = |𝒁𝟎| + |𝒁𝟏| + |𝒁𝟐| + ⋯+ |𝒁𝑵|
E uma vez enquanto um Ômega Eugênio Número. Suponho, por princípio, que:
𝑴(∞)𝝎𝑩(∞)𝝎 ≤ 𝜷𝝎𝑳(𝑷) < 𝑴(∞)𝑹𝒆𝒅𝑩(∞)𝑹𝒆𝒅 ∴ (𝜷𝝎 ∈ 𝑸+
∗ )
Ômega varia entre Trans100dente e Algébrico até corroborações definitivas ao seu Status.
9. 𝝎𝒌
= 𝟐𝑪 +
𝟏
𝝎𝒌
∴ 𝝎 = √√𝑪𝟐 + 𝟏 + 𝑪
𝒌
∴ ℵ = 𝑭(√𝟐𝑪
𝒌
)
𝝎𝒌
= 𝑳𝒊𝒎𝒏→∞〈𝟐𝟏𝑪; 𝟐𝟐𝑪,𝟐𝟑𝑪, 𝟐𝟒𝑪,⋯ , 𝟐𝒏𝑪〉 ≜ [𝟐𝑪]
Um Ômega que pela teoria dos números sendo um Algébrico Irracional
ou um Transcendente. Terá, pela minha Téos Orgia numérica, a depender
da quantidade dos seus dígitos, um comportamento muito mais voltado
ao Algébrico e em outro momento para o Trans100dente. Logo é possível
a raiz quadrada de um primo em duzentos mil dígitos comportando como
um Trans100dente ao ter o grau de Trans100dência maior ou igual a um.
Mas quando e a partir dos setecentos mil dígitos em um comportamento
definitivamente Algébrico com o grau de Trans100dência menor que um.
Todo Eugênio transita de Trans100dente a Algébrico por simplesmente
ir variando adequadamente a quantidade dos seus dígitos. Existindo uma
Transcendência Aritméticas tanto como uma Trans100dência Numéricas.
12. Olimpíadas Almejado
Gens de Infinito Dígitos
Dado um conjunto de Eugênio Números Geneticamente múltiplos de X. Oriundos de um monômio
crescente com coeficiente maior que zero. Dos quais determina-se os respectivos Eugênio Zeus [X].
Tal como neste Eugênio Triplo:
(𝟏, 𝒁)𝑬𝒖𝒈 [𝑿] ∴ 𝑬𝒖𝒈 = {𝑬𝑵 = 𝑿|𝑷(𝑵)|
⃡ ∴ 𝑵 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, ⋯ }
𝑷(𝑵) = 𝑹𝑵𝒒
∴ 𝑷(𝑵 + 𝟏) > 𝑷(𝑵) ∴ 𝒒 ∈ {𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎, ⋯}
𝑻𝒓𝒊𝒑𝒍𝒐 = {𝑻𝑵 = 𝟑𝑵 ∴ 𝑵 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎, 𝟏𝟏, 𝟏𝟐, 𝟏𝟑, 𝟏𝟒, ⋯ }
Encontrando o maior múltiplo de três nessa sucessão infinita de múltiplos de três:
(𝟏, 𝟎)𝑩𝒊𝒂 = (𝟏, 𝟐)𝑻𝒓𝒊𝒑𝒍𝒐[𝟑] = (𝟑𝟔𝟗), (𝟏𝟐)𝟏𝟓𝟏𝟖𝟐𝟏𝟐𝟒(𝟐𝟕𝟑𝟎𝟑𝟑𝟑𝟔𝟑𝟗)⋯
Veja que, assim como (1, Zero) Zeus, todos tem seu Infinito Partição na última posição:
(𝟏, 𝟎)𝒁𝒆𝒖𝒔 = 𝟐, 𝟑𝟓𝟕(𝟏𝟏)𝟏𝟑𝟏𝟕𝟏𝟗𝟐𝟑𝟐𝟗𝟑𝟏𝟑𝟕𝟒𝟏𝟒𝟑𝟒𝟕𝟓𝟑⋯(𝑫𝜹𝑫𝜹+𝟏 ⋯ 𝑫∞)
havendo infinitas partições quase infinito primo pelo Zeus particionamentos Clássicos:
(𝟏, 𝟏)𝒁𝒆𝒖𝒔 [𝑪] = (𝟐𝟑), (𝟓)(𝟕𝟏)(𝟏𝟏𝟑)(𝟏𝟕)𝟏𝟗𝟐𝟑(𝟐𝟗𝟑) ⋯ (𝑫𝜽𝑫𝜽+𝟏 ⋯ 𝑫∞−𝒌)⋯
Sendo que de cinco em cinco Ãnus e com esses Eugênios. A Universidade que estiver com a partição
mais próxima a infinita vence a olimpíada. Para que tenhamos o fator sorte, os experimentos serão
executados às cegas para a Universidade não ter conhecimento de quem é o seu número olímpico.
Então e supondo que a Universidade Federal da Bahia consiga encontrar uma partição de vinte mil
dígitos com as demais no máximo conseguindo só os onze mil. Portanto será a vencedora ganhando
o prêmio referente ao quinquênio olimpíada anterior a atual. Permanecendo o experimento até o
limite de operacionalidade eletrônica dessa máquina de cálculos utilizada exclusivamente para tal.
![Ômega Eugênio Zeus [XCor] Krishna
Eugenio Evangelista Souza. Unir Versus Ãbertas
Téos Orgias Zeus:
Eugênio Números Alpha
±(𝟎, 𝟎)𝑬𝒖𝒈 = 𝟎 ∴ ±(𝟎, 𝑴 > 𝟎)𝑬𝒖𝒈 = ±(𝟏, 𝑴 − 𝟏)𝑬𝒖𝒈
(±𝑵, ±𝑴)𝑬𝒖𝒈 = (𝑵, 𝑴)𝑬𝒖𝒈 ∴ (±𝑵, ∓𝑴)𝑬𝒖𝒈 = −(𝑵, 𝑴)𝑬𝒖𝒈
±(𝑵, 𝑴)𝑬𝒖𝒈 = ±𝑬𝑵𝑬𝑵+𝟏𝑬𝑵+𝟐 ⋯ 𝑬𝑵+𝑴,𝑬𝑵+𝑴+𝟏𝑬𝑵+𝑴+𝟐𝑬𝑵+𝑴+𝟑 ⋯
𝑬𝒖𝒈 = {𝑬𝑵 = |𝑮(𝑵)|
⃡ ∈ 𝒁+ ∴ 𝑵 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎, 𝟏𝟏, 𝟏𝟐, ⋯}
|𝑮(𝑵)|
⃡ = Parte Inteira do Valor Absoluto (Ou não Negativo) da função G de N.
𝑲𝒓𝒊𝒔𝒉𝒏𝒂 = {𝑲𝑵 = 𝑵 ∴ 𝑵 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, ⋯ , 𝟐𝟐, 𝟐𝟑, 𝟐𝟒, 𝟐𝟓, ⋯}
−(𝟐𝟐, 𝟑)𝑲𝒓𝒔𝒏
⏟𝒂 = −(𝟐𝟐)𝟐𝟑𝟐𝟒𝟐𝟓, 𝟐𝟔𝟐𝟕𝟐𝟖𝟐𝟗𝟑𝟎 ⋯ 𝟗𝟕𝟗𝟖𝟗𝟗(𝟏𝟎𝟎)𝟏𝟎𝟏𝟏𝟎𝟐 ⋯
Particionamentos em Gens Numéricos Naturais
[𝑬𝑵]𝑫𝒈𝒕 = 𝟏 ↔ 𝑬𝑵 ∴ [𝑬𝑵+𝑺]𝑫𝒈𝒕 = [𝑬𝑵+𝑺−𝟏]𝑫𝒈𝒕 ↔ 𝑬𝑵+𝑺 ∴ {𝑬𝑵 = (𝑵, 𝟎)𝑬𝒖𝒈
⃡ }
[𝑬𝑵]𝑫𝒈𝒕 > 𝟏 ↔ (𝑬𝑵) ∴ [𝑬𝑵+𝑺]𝑫𝒈𝒕 ≠ [𝑬𝑵+𝑺−𝟏]𝑫𝒈𝒕 ↔ (𝑬𝑵+𝑺) ∴ [𝟐𝟏𝟒]𝑫𝒈𝒕 = 𝟑 dígitos
Eugênio Números Beta Zeus [2Azul] Particionados
(𝟏, 𝟎)𝑻𝒆𝒔𝒂𝒐 = (𝑬𝒖𝒍𝒆𝒓)𝑬𝒖𝒈[𝟐] = 𝟐, (𝟕𝟏𝟖𝟐𝟖)𝟏𝟖𝟐𝟖𝟒(𝟓𝟗𝟎𝟒)(𝟓𝟐)(𝟑𝟓𝟑𝟔𝟎𝟐𝟖)(𝟕𝟒) ⋯
𝟐(𝑷𝒂𝒓) ∴ 𝟐𝟕(𝑰𝒎𝒑𝒂𝒓) = 𝟐(𝟕 ⋯ ) ∴ 𝟕, 𝟕𝟏(𝑰𝒎𝒑𝒂𝒓𝒆𝒔)𝟕𝟏𝟖, 𝟕𝟏𝟖𝟐, 𝟕𝟏𝟖𝟐𝟖(𝑷𝒂𝒓𝒆𝒔) ∴
𝟕𝟏𝟖𝟐𝟖𝟏(𝑰𝒎𝒑𝒂𝒓) = (𝟕𝟏𝟖𝟐𝟖)(𝟏⋯ ) ∴ 𝟏(𝑰𝒎𝒑𝒂𝒓)𝟏𝟖, 𝟏𝟖𝟐, 𝟏𝟖𝟐𝟖, 𝟏𝟖𝟐𝟖𝟒(𝑷𝒂𝒓𝒆𝒔) ∴
𝟏𝟖𝟐𝟖𝟒𝟓(𝑰𝒎𝒑𝒂𝒓) = 𝟏𝟖𝟐𝟖𝟒(𝟓 ⋯) ∴ 𝟓, 𝟓𝟗(𝑰𝒎𝒑𝒂𝒓𝒆𝒔)𝟓𝟗𝟎, 𝟓𝟗𝟎𝟒(𝑷𝒂𝒓𝒆𝒔) ∴ (⋯ ) ⋯](https://image.slidesharecdn.com/palestra-210926200731/85/Teoria-Numerica-Palestra-01-1-320.jpg)
![(𝟏, 𝟎)𝑹𝒆𝒅 = (𝟏, 𝟏)𝑲𝒓𝒔𝒏
⏟𝒂[𝟐] = (𝟏𝟐), 𝟑𝟒𝟓𝟔𝟕𝟖(𝟗𝟏𝟎)(𝟏𝟏𝟏𝟐)𝟏𝟑𝟏𝟒𝟏𝟓𝟏𝟔𝟏𝟕𝟏𝟖(𝟏𝟗𝟐𝟎𝟐) ⋯
(𝟏, 𝟎)𝑻𝒉𝒐𝒓 = (𝟏, 𝟏)𝒁𝒆𝒖𝒔[𝑪𝒍𝒂𝒔𝒔𝒊𝒄𝒐] = (𝟐𝟑), (𝟓)(𝟕𝟏)(𝟏𝟏𝟑)(𝟏𝟕)𝟏𝟗𝟐𝟑(𝟐𝟗𝟑)𝟏𝟑𝟕(𝟒𝟏)𝟒𝟑 ⋯
𝝋 = 𝝋𝟎𝝋𝟏 ⋯ 𝝋𝒀, 𝝋𝒀+𝟏𝝋𝒀+𝟐 ⋯ 𝝋𝑺−𝟏 ⋯ ∴ 𝝋 = 𝟎, 𝟎 ⋯ 𝟎𝝋𝟎𝝋𝟏 ⋯ 𝝋𝑺−𝟏 ⋯ ∴ 𝝋𝟎
≠ 𝟎
𝝋𝒒 ∈ {𝟎, 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗} ∴ ⟨𝝎|𝑺|𝑹𝒆𝒅⟩ = (𝝋𝟎 ⋯ 𝝋𝜶) ⋯ (∴)(⟦𝑸(𝑺)⟧)(𝝋𝜹 ⋯ 𝝋𝑺−𝟏
(𝟏, 𝟎)𝑬𝒖𝒈[𝑳, 𝟏, −𝑲]𝑷−𝟏 = 𝑬𝟏𝑳𝟎
+ 𝑬𝟐𝑳𝑮−𝟏 + 𝑬𝟑𝑳𝑮−𝟐 + 𝑬𝟒𝑳𝑮−𝟑 + ⋯+ 𝑬𝑲+𝟐𝑳𝑮−𝑲−𝟏
𝑮𝟎 = 𝟎 ∴ 𝑮−𝑯 = 𝑮𝟏−𝑯 − [𝑬𝑯+𝟏]𝑫𝒈𝒕 ∴ 𝑯 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, ⋯ , 𝑸(𝑺) − 𝟐, (𝑸(𝑺) − 𝟏)
𝝑[(𝟏, 𝟎)𝑬𝒖𝒈[𝟏𝟎, 𝟏, −𝑲]𝑷−𝟏;𝑻]
𝑷
= {𝑬𝟏𝑻𝟎(𝟎)/,/𝑬𝟐𝑻𝑮−𝟏(𝑮−𝟏)/ ⋯/𝑬𝑲+𝟐𝑻𝑮−𝑲−𝟏(𝑮−𝑲−𝟏)}
𝝑
̂[𝟏]𝒈(𝜽)
±
= √
∑(𝑮𝟏−𝑯 − 𝑮−𝑯)𝟐
(𝑲 + 𝟐) + 𝟐 − 𝟏
∴ ℵ𝑲
[
𝝑[(𝟏, 𝟎)𝑬𝒖𝒈[𝟏𝟎,𝟏, −𝑲]𝑷−𝟏;𝟐−𝟏]
𝑷
𝑳𝒐𝒈𝟐 (
𝝑
̅[𝟐]𝒈(𝜽)
±
𝑩(𝑼)𝟐𝑴(𝑼)
)[
(𝑹𝒆𝒅)(𝝎𝑬𝒖𝒈)
𝑲 = 𝟏,𝟐, ⋯ , ∆
]
]
{𝝑
̅[𝟐]𝒈(𝜽)
±
=
∑ 𝑬𝑯𝟐−𝑮𝟏−𝑯
(𝑲 + 𝟐)𝝑
̂[𝟏]𝒈(𝜽)
±
+ 𝟐
} ∴ {
(𝟏, 𝟎)𝑹𝒆𝒅 = (𝟏, 𝒁)𝑲𝒓𝒔𝒏
⏟𝒂[𝑿]
𝑿 ∈ {𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, ⋯ }
}
𝑴(𝑼) = [(𝑩(𝑼), 𝟎)𝑬𝒖𝒈
⃡ ]
𝑫𝒈𝒕
> [(𝑿, 𝟎)𝑬𝒖𝒈
⃡ ]
𝑫𝒈𝒕
∴ ∀𝑿 < 𝑩(𝑼) ≤ 𝑩(∞) ≤ 𝑸(𝑺) < 𝑺
𝑴(𝟐) > 𝑴(𝟏) = [(𝟐, 𝟎)𝑻𝒆𝒔𝒂𝒐
⃡ ]
𝑫𝒈𝒕
= [𝟕𝟏𝟖𝟐𝟖]𝑫𝒈𝒕 > 𝟏 = [(𝟏, 𝟎)𝑻𝒆𝒔𝒂𝒐
⃡ ]
𝑫𝒈𝒕
= [𝟐]𝑫𝒈𝒕
𝑴(𝟑) = [(𝟖, 𝟎)𝑻𝒆𝒔𝒂𝒐
⃡ ]
𝑫𝒈𝒕
= [𝟕𝟏𝟑𝟓𝟐𝟔𝟔𝟐𝟒]𝑫𝒈𝒕 = 𝟗 > 𝟕 = [(𝟔, 𝟎)𝑻𝒆𝒔𝒂𝒐
⃡ ]
𝑫𝒈𝒕
= 𝑴(𝟐)
𝑴(∞) > ⋯𝑴(𝟒) = [(𝟔𝟓, 𝟎)𝑻𝒆𝒔𝒂𝒐
⃡ ]
𝑫𝒈𝒕
= [𝟏𝟒𝟔𝟎𝟔𝟔𝟖𝟎𝟖𝟐𝟐𝟔𝟒𝟖𝟎𝟎]𝑫𝒈𝒕 = 𝟏𝟔 > 𝑴(𝟑)
{
.
.
→ (𝝎)𝑰𝒓𝒓𝒂𝒄𝒊𝒐𝒏𝒂𝒍(∞ ≥ 𝑺) ↓
∆ = 𝑩(∞ ≫ 𝟐)𝑴𝒊𝒏 − 𝟐 ≫ 𝟏 →
→
𝑮𝒓𝒂𝒇𝒊𝒄𝒐𝒔
(𝑿𝑪𝒐𝒓)[∆]
[∆
̃$(∞) =
ℵ∆(𝑹𝒆𝒅)
ℵ∆(𝝎)
+
𝑸(∞)𝝎
𝑸(∞)𝑹𝒆𝒅
]
.
.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)