Can you construct a regular pentadecagon (15-sided) polygon with compass and straightedge? What’s the restriction on n≥3 such that a regular n-gon is constructible?

Constructing an equilateral triangle, n=3, is very simple, as are constructing a square, n=4, and a hexagon, n=6. The method of constructing a regular pentagon, n=5, has been known since the days of Euclid.
In addition, if one can construct a regular n-gon, then by bisecting its sides one can construct a 2n-gon; by repeating, the polygons are thus constructible for 4n, 8n, etc.

Further, suppose that we can construct a regular n-gon and d≥3 is a divisor of n. Then by connecting every vertices of the n-gon, one produces a d-gon. So, suppose that the regular k-gon is not constructible. Then we can conclude that a polygon with a number of sides that is any multiple of k is also not constructible.

Now, suppose we have that a regular a-gon is constructible, and the regular b-gon is also constructible, with a and b coprime (gcd(a,b)=1). Suppose we construct these two polygons inscribed in the same circle with one (and only one, since a and b are coprime) vertex in common. Then imagine the regular polygon with ab sides. One can form a regular a-gon by connecting every b vertices, and a regular b-gon by connecting every a vertices. Now, Bézout’s identity tells us that for any two non-zero integers a and b, there exist integers u and v such that , which means that for our coprime a and b, there exist integers u and v for which , and thus there exist integer multiples of a and b which differ by exactly one. This means that in our construction of the a-gon and b-gon in the same circle with common vertex, there exist a vertex of the a-gon and a vertex of the b-gon which form one side of the ab-gon, and from that side and the circle in which the polygons are inscribed, one can construct the entire ab-gon. Thus, if the regular a-gon and regular b-gon are both constructible with a and b coprime, then the regular ab-gon is constructible.

So since 15=3*5, and the equilateral triangle and pentagon are both constructible, the regular pentadecagon is thus constructible. To use this as an example of the argument above, we have that 3*2+5*(-1)=1. Thus, traveling around from the common vertex, if we take two sides of the pentagon and one side of the triangle, the resulting vertices are the sixth and fifth around of the pentadecagon, respectively, and so they form one side of the pentadecagon.



The question of what values of n allow the regular n-gon to be constructible was studied by many mathematicians; Gauss developed a sufficient condition in 1796 which he conjectured was also necessary; Pierre Wantzel later proved that it was so. The proof arises from analytic geometry, and is best expressed in terms of field theory and, in particular, quadratic extensions. The result, however, is that the regular n-gon is constructible if and only if φ(n) is a power of 2, where φ(n) is Euler’s totient function. Our result about ab with coprime a and b thus follows from φ(n) being a multiplicative function.
Since for prime p, we see that , again confirming that if n gives us a constructible polygon, n times any power of two also gives us a constructible polygon. As for prime p>2, we see that is divisible by p for k>1, and is thus not a power of 2 in that case; so, no odd prime may appear as a factor with multiplicity greater than 1; thus the enneagon, , is not constructible. Further, a prime factor p will be able to give a power of 2 only if is a power of 2, which means that p is of the form . I proved last week that the only primes of this form are the Fermat primes, primes of the form . Thus, the above is equivalent to Gauss’ phrasing that a regular n-gon is constructible with compass and straightedge if (and only if) n is the product of a power of 2 (including 2⁰=1) and any number (including zero) of distinct Fermat primes.
So we see that the first few constructible regular polygons are
n=3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, …
while the first few non-constructible regular polygons are
n=7, 9, 11, 13, 14, 18, 19, …

Since the only known Fermat primes are F₀=3, F₁=5, F₂=17, F₃=257, and F₄=65537, their product, 4,294,967,295, is thus the largest odd number of sides for which the regular polygon is known to be constructible.

Einstein speaks on General Relativity; (Nov. 20, 2009)

I have already posted two articles in the series “Einstein speaks on…” This article describes Einstein’s theory of restricted relativity and then his concept for General Relativity. It is a theory meant to extend physics of fields (for example electrical and magnetic fields among others) to all natural phenomena, including gravity. Einstein declares that there was nothing speculative in his theory but it was adapted to observed facts.

The fundamentals are that the speed of light is constant in the void and that all systems of inertia are equally valid (each system of inertia has its own metric time). The experience of Michelson has demonstrated these fundamentals. The theory of restrained relativity adopts the continuum of space coordinates and time as absolute since they are measured by clocks and rigid bodies with a twist: the coordinates become relative because they depend on the movement of the selected system of inertia.

The theory of General Relativity is based on the verified numerical correspondence of inertia mass and weight. This discovery is obtained when coordinates posses relative accelerations with one another; thus each system of inertia has its own field of gravitation. Consequently, the movement of solid bodies does not correspond to the Euclidian geometry as well as the movement of clocks. The coordinates of space-time are no longer independent. This new kind of metrics existed mathematically thanks to the works of Gauss and Riemann.

Ernst Mach realized that classical mechanics movement is described without reference to the causes; thus, there are no movements but those in relation to other movements.  In this case, acceleration in classical mechanics can no longer be conceived with relative movement; Newton had to imagine a physical space where acceleration would exist and he logically announced an absolute space that did not satisfy Newton but that worked for two centuries. Mach tried to modify the equations so that they could be used in reference to a space represented by the other bodies under study.  Mach’s attempts failed in regard of the scientific knowledge of his time.

We know that space is influenced by the surrounding bodies and so far, I cannot think the general Relativity may surmount satisfactorily this difficulty except by considering space as a closed one assuming that the average density of maters in the universe has a finite value, however small it might be.

Einstein speaks of his mind processes on the origin of General Relativity; (Nov. 21, 2009)

This article is how Einstein described his mind processes that lead to the theory of restricted relativity and then his concept for General Relativity. In 1905, restricted relativity discovered the equivalence of all systems of inertia for formulating physics equations. From a cinematic perspective there was no way to doubting relative movements; still there was the tendency to physically extend privileged significance to system of inertia.  The question was “if speed is relative do we have to consider acceleration as absolute?”

Ernest Mach considered that inertia did not resist acceleration except when related to the acceleration toward other masses. This idea impressed me greatly. First, I had to establish a law of gravitation field and suppress the concept of absolute simultaneity. Simplicity urged me to maintain Laplace’s scalar gravity potential and fine tune Poisson’s equation. Given the theorem of inertia of energy then inertia mass must be depended on gravitation potential but my research left me skeptical. In classical mechanics, vertical acceleration in a vertical field of gravity is independent of the horizontal component of velocity; it follows that vertical acceleration is exercised independently of the internal kinetic energy of the body in movement. I discovered that this independence did not exist in my draft theory; this evidence did not coincide with the affirmation that all bodies submit to the same acceleration in a gravitational field. Thus, the principle that there is equality between inertia mass and weight grew with striking significance. I was convinced of its validity though I had no knowledge of the results of experiments done by Eotvos.

Consequently, the principle of equality between inertia mass and weight would be explained as follow: in a homogenous gravitational field all movements are executed in relation to a system of coordinates accelerating uniformly as if in absence of gravity field. I conjectured that if this principle is applicable to any other events then it can be applied to system of coordinates not accelerating uniformly. These reflections occupied me from 1908 to 1911 and I figured that the principle of relativity needed to be extended (equations should retain their forms in non uniform accelerations of coordinates) in order to account for a rational theory of gravitation; the physical explanation of coordinates (measured by rules and clocks) has to go.

I reasoned that if in reality “a field of gravitation used in system of inertia” did not exist it could still be served in the Galilean expression that “a material point in a 4-dimentional space is represented by the shortest straight line”. Minkowski has demonstrated that this metric of the square of the distance of the line is a function of the squares of the differential coordinates.  If I introduced other coordinates by non linear transformation then the distance of the line stay homogeneous if coefficients dependent on coordinates are added to the metric (this is the Riemann metric in 4-dimension space not submitted to any gravity field). Thus, the coefficients describe the field of gravity in the selected system of coordinates; the physical significance is just related to the Riemannian metric. This resolved this dilemma in 1912.

Two other problems had to be resolved from 1912 to 1914 with the collaboration of Marcel Grossmann. The first problem is stated as follow: How can we transfer to a Riemannian metric a field law expressed in the language of restrained relativity?  I discovered that Ricci and Levi-Civia had answered it using infinitesimal differential calculus.  The second problem is: what are the differential laws that determine the coefficients of Riemann?  I needed to resolve invariant differential forms of the second order of Riemann’s coefficients. It turned out that Riemann had also answered the problem using curb tensors.

“Two years before the publication of my theory on General Relativity” said Einstein “I thought that my equations could not be confirmed by experiments. I was convinced that an invariant law of gravitation relative to any transformations of coordinates was not compatible with the causality principle. Astronomic experiments proved me right in 1915.”

Note:  I recall that during my last year in high school my physics teacher, an old Jesuit Brother, filled the blackboard with partial derivatives of Newton’s equation on the force applied to a mass; then he integrated and he got Einstein’s equation of energy equal mass by C square. At university, whenever I had problems to solve in classical mechanics on energy or momentum conservations I just applied the relativity equation for easy and quick results; pretty straightforward; not like the huge pain of describing or analyzing movements of an object in coordinate space.

NORMAL
DEX98: Care este așa cum trebuie să fie, potrivit cu starea firească, obișnuit, firesc, natural

NODEX02: Care este în concordanță cu norma; conform normei; firesc; natural; obișnuit.
2) (despre ființe vii) Care este în conformitate cu normele speciei sale (din punct de vedere fizic și psihic)

* observati schimbarea sensului cuvantului in decursul a patru ani – normal este acum mai mult sinonim cu “standard”, adica o conventie. Inainte avea un sens mai “absolut”, adica, anormalul era o “abominatie”, iar acum anormalul este doar “neconform unei conventii”.

ANORMAL
Dex98: Care se abate de la normal; contrar regulilor, normelor obișnuite.
Antonime 2002: nebun, nenormal, patologic, subnormal
Dictionary.com: care nu e normal, obisnuit, des intalnit, comun, care se departeaza de medie, de o norma sau un sablon.

* observati diferenta de perceptie intre natiuni asupra anormalului. Anormalul romanesc este “bolnav”, iar cel american este mai mult “inedit” sau “neasteptat”.

NÓRMĂ ~e f.
1) Sumă de reguli recunoscute de majoritatea unei colectivități ca obligatorii.
2) Criteriu de apreciere
3) Mărime medie admisă pentru ceva; măsură stabilită.

Normalitatea in orice context uman reprezinta o medie a unui intreg, un punct ce asigura o usurinta de comunicare, cooperare si identificare si o reflexie asupra mediului in sine in care se manifesta un grup, necesitatile implinite si neimplinite ale fiintei umane intr-un spatiu si un moment din timp.
Odata cu migrarea acestui intreg si transformarea acestuia in functie de mediul in care se afla, ceea ce este considerat normal, la fel, se se adapteaza si este adaptat, pentru a intruni necesitatile majoritatii.
Normalul este firescul creat de om si totodata trait de animalul ce-l reprezinta.
Omul in sine, ca ideal, isi schimba identitatea si definitia de la o epoca la alta, de la o zona la alta, chiar de la individ la individ.
Cooperarea si comunicarea sunt necesare omului social pentru a supravietui.
Prioritatile si normele din viata acestuia sunt stabilite de nevoile acestuia.
Ne dorim haos in contextul in care dorim evolutie si un statut. Ne dorim ordine in contextul in care dorim siguranta.
Normele ne docilizeaza si in acelasi timp ne revolta. Le recunoastem ca un firesc in viata noastra, si totusi le respingem absurditatea si inutilitatea pentru persoana proprie.
E normal sa dam noroc, sa dam buna ziua cunoscutilor, sa tinem la aspectul propriu si la visele noastre.
Si cu toate astea ni se pare anormal sa nu ne putem exprima adevaratele opinii in fata autoritatilor din viata noastra, ne deranjeaza sa observam aparenta uniformizare pe strada, care nu ne caracterizeaza. Discriminam si nu vrem sa fim discriminati, deoarece nu putem accepta realitatea existentei tuturor din jurul nostru, care sunt altfel decat noi.
Admiram oamenii cu personalitate, dar doar pe cei pe care ii putem intelege. Detestam oamenii cu un caracter bazat pe alte valori decat ale noastre.
Traim intr-o lume atat de colorata, si totusi tindem sa apreciem doar putinele elemente care ne intereseaza, pentru ca varietatea “buna” din exterior se rezuma la ceea ce necesitam in interior.
Ne simtim luminati ca adoptam noi curente. Si cu toate astea, ele sunt adoptate de majoritate. Ceea ce ne face speciali prin a fi la fel.
Ne simtim avangardisti cand imbracam cea mai noua haina sau asimilam cea mai noua idee… Si cu toate acestea, ideile acestea pot fi calculate, prezise si deduse logic. Tendintele exista… Iar orice tendinta prezentata prea devreme ni se pare o abominatie. Insa la momentul potrivit, tendinta este un lucru minunat, pe care, daca il adoptam, ne hraneste egoul si ne da impresia ca suntem CINEVA, presus CELORLALTI.
Contraculturile si subculturile ingrozesc, deoarece isi situeaza normalitatea proprie departe de limitele normale ale culturii in care se afla.
Si cu toate acestea, in cadrul contraculturii, membrii lor sunt conformisti. Formeaza omogenitate si o identitate colectiva. Poate chiar o omogenitate mai mare decat grupul din care s-au debarasat. Deoarece un grup, cu cat este mai nou, cu atat este mai coerent si uniform, in timp dezbinandu-se entropic, pana la uitare. Ca apoi sa fie reinventat si adaptat nevoilor din alta perioada…

Si ce ne face sa fim Cineva? In ce context? Suntem cineva pentru cei mai apropiati din grupul in care ne aflam. Suntem nimeni, sau doar o statistica in societate.
Suntem cineva in fata viselor pe care le urmam si care ne aduc bucuria de a trai. Suntem nimeni in fata mortii.

Ce este normalitatea? Nimic altceva decat o iluzie, o conventie temporara si dinamica in timp. Ea evolueaza, se schimba odata cu lumea si cu nevoile ei. Limitele ei sunt mai largi sau mai stranse in functie de context. Este punctul de sprijin si de orientare pentru oamenii care se misca permanent, care se schimba permanent, biologic, psihologic, intelectual.

Ce este anormalitatea? Iesirea din acele limite ce permit individului sa fie acceptat si dezirabil in societatea sau grupul in care se afla. Anormalul trece dincolo de limitele acceptate ale normalului, el socheaza, dezgusta si creeaza un sentiment de haos individului. El poate fi inspaimantator, sau demn de dispret. Poate trezi ura si resentimente. Este lucrul pe care oricine ar dori sa-l elimine din existenta si sa-l expulzeze din perceptia sa.

Calitatile acceptate intr-un sistem de valori, daca sunt exacerbate, devin parte a anormalului pentru individul normal, asadar generand un sentiment de respingere.”Geniul” ii va parea omului normal de neinteles, intimidant si ii va crea un dezechilibru interior, care ii va nega necesitatea naturala de a fi valid. Astfel el va deveni opac in fata acestuia, pentru a-si proteja sistemul de valori.

Care sunt limitele normalitatii? Ele nu pot fi definite decat in contexte, intr-un moment din timp. Nu exista un algoritm care sa poata calcula aceste limite in functie de grup. Insa ele pot fi calculate in cadrul aceluiasi grup, ca evolutie. Cum putem realiza dimensiunile acestor limite cand nu ne realizam nici macar propriile limite?

Si ce reprezinta depasirea limitelor altceva decat largirea limitelor intre care se incadreaza normalitatea? A depasi limite duce la anormalitate. A te juca pe marginile limitelor inseamna a-ti manifesta in societate maximul acceptabil si traductibil din propria personalitate, fara sa fii marginalizat pentru transparenta prin care te expui.

In aceasta lumina putem accepta ca limitele normalitatii nu sunt decat o norma sociala, urmare a tendintei de uniformitate intrinseca oricarei forme de organizare umane, norme dictate de segmentul mai numeros al populatiei. De-a lungul istoriei aceste “limite” au suferit o serie de transformari, in functie de contextul psiho-social al diverselor epoci. Astfel sfintii au devenit nebuni, nebunii au devenit oameni de stiinta sau arte, dar fenomenul in sine nu s-a schimbat.

Es la pregunta que se hace Acebron, el autor de Tall&Cute, y que intenta responder en este interesante artículo:

Una de curiosidades matemáticas más comentadas es el denominado último teorema de Fermat, un problema matemático con una historia poco usual y seguro conocida por muchos lectores.

El gran genio francés acostumbraba a escribir anotaciones en el borde de los libros que leía aunque es especialmente famosa su anotación realizada en 1637 en un ejemplar de La Aritmética de Diofanto , que asegura… Seguir leyendo

Dalam menyusun SRF, banyak metode yang digunakan salah satunya metode kuadrat terkecil biasa. Metode ini pertama kali dikemukkan oleh Carl Friedrich Gauss, seorang ahli matematika dari Jerman.

Metode kuadrat terkecil (OLS) atau metode kemungkinan maksimum (method of maximum likehood) adalah salah satu metoda yang paling populer dalam menyelesaikan masalah hitung perataan.

Metode kuadrat terkecil biasa dikemukakan oleh Carl Friedrich Gauss. Gauss membuat asumsi-asumsi sebagai berikut.

Asumsi 1

E(u_i‌‌‌‌‌‌| X_i) = 0

Artinya nilai yang diharapkan bersyarat (conditional expected value) dari u_i, tergantung pada X_i tertentu, adalah nol.

Asumsi 2

cov(u_i , u_j) = E[u_i – E(u_i)][u_j – E(u_j)]

= E(u_i , u_j)       karena asumsi 1

= 0                   i ≠ j

Dimana i dan j dua pengamatan yang berbeda dan dimana cov berarti kovarians.

Hal itu mendalilkan bahwa gangguan u_i dan u_j tidak berkorelasi. Asumsi ini dikenal sebagai asumsi tidak adanya korelasi berurutan atau tidak ada autokorelasi.

Asumsi 3

var(u_i | X_i)        = E[u_i – E(u_i)]²

= E(u_i²)            karena asumsi 1

= σ²

var = varians

Persamaan di atas menyatakan bahwa varians u_i untuk tiap X_i (yaitu varians bersyarat untuk u_i) adalah suatu angka konstan positif yang sama dengan σ². Bisa dikatakan juga bahwa populasi Y yang berhubungan dengan berbagai nilai X mempunyai varians yang sama.

Jika varians tidak sama, atau sering disebut heteroskedastisitas, maka varians bersyarat populasi Y meningkat bersama dengan meningkatnya X.

var (u_i| X_i) = σ_i²

Asumsi 4

cov (u_i, X_i)       = E[u_i – E(u_i)][X_i – E(X_i)]

= 0

Menyatakan bahwa gangguan u dan variabel yang menjelaskan X tidak berkorelasi. Asumsi 4 dapat terpenuhi jika variabel X tidak random atau tak stokhastik.

Asumsi model regresi linier klasik

No asumsi                   Dalam hubungannya               Dalam hubungannya

dengan u dengan Y

1. E(u_i| X_i) = 0                             E(Y_i| X_i) = β₀ +β₁ X₁
2. cov(u_i|u_j) = 0     i≠j                  E(Y_i| Y_j) = 0     i≠j
3. var(u_i| X_i) = σ² var(Y_i| X_i) = σ²

dapat dikatakan bahwa asumsi Gauss menunjukkan bahwa penaksir parameter regresi yang diperoleh dengan metode kuadrat terkecil adalah optimum.

Prinsip Kuadrat Terkecil

e_i = Y_i – Ŷ_i

= Y_i – β₀ – β₁ X_i

Menunjukkan bahwa e_i (residual) hanyalah perbedaan antara nilai Y sebenarnya dengan yang ditaksir.

Jika menggunakan kriteria   ∑ e_i = (Y_i – Ŷ_i) maka semua residual menerima (tingkat) penting yang sama, tidak peduli seberapa dekat atau seberapa jauh terpencarnya observasi individual dari SRF. Jumlah residual ini nol.

Kita bisa menggunakan kriteria kuadrat terkecil untuk manghindari jumlah residual nol, yaitu sebagai berikut.

∑ e_i = (Y_i – Ŷ_i)²

=(Y_i – β₀ – β₁ X_i)²

Tetapi hal ini tak mungkin karena semakin besar ei (dalam nilai mutlak), semakin besar ∑ei.

Pembenaran lebih lanjut, kita gunakan penaksir kuadrat terkecil.

∑ e_i² = f ( β₀. β₁)

Jumlah residual/ sisa kuadrat adalah suatu fungsi dari penaksir β₀ dan β_1. dengan memeilih nilai yang berbeda akan memberikan e yang berbeda dan karenanya nilai ∑ei2 yang berbeda.

Persamaan normal, dimana N adalah besarnya sampel. Dinyatakan:

∑Yi = N β₀ + β₁ ∑Xi

∑YiXi = β_0Xi + β₁ ∑Xi2

β₀₌ Ŷ – β₁X

Ciri-ciri penaksir kuadrat terkecil:

1. Penaksir tadi dinyatakan semata-semata dalam bearan yang bisa diamati (besaran sampel)
2. Penaksir tadi merupakan penaksir titik (point estimator) yaitu dengan sampel tertentu, tiap penaksir akan memberikan hanya satu nilai (ttiik) tunggal parameter populasi yang relevan.

Garis regresi yang diperoleh demikian mempunyai sifat-sifat berikut.

1. Garis regresi melaui rata-rata sampel X dan Y.
2. Nilai rata-rata Y yang ditaksir (=Ŷi) adalah sama dengan nilai rata-rata Y yang sebenarnya.
3. Nilai rata-rata residual ei adalah nol.
4. Residual ei tak berkorelasi dengan Yi yang ditaksir.
5. Residual ei tak berkorelasi dengan Xi

Ketepatan (presisi) atau kesalahan Standar Taksiran Kuadrat Terkecil

Dalam statistik ketepatan taksiran diukur dengan kesalahan standar (se). Kesalahan standar taksiran OLS dapat diperoleh sebagai berikut.

Var (β1) =

Se (β1) =

Var (β0) =

Se (β0) =

Sifat-sifat Penaksir Kuadrat Terkecil

Teorema Gauss – Markov

Koefisien Determinasi r² – Suatu Ukuran Kebaikan – Suai (Goodness of Fit)

Koefisien determinasi (r²) merupakan suatu ukuran yang menyatakan seberapa baik garis regresi sampel mencocokkan data.

Untuk menghitung r² dalam bentuk simpangan:  y_i = ŷ_i = e_i

Jika dikuadratkan pada kedua sisis dan menjumlahkan untuk semua sampel, diperoleh:

∑yi2 = ∑ŷi2 + ∑ei2 + 2 ∑ŷiei

= ∑ŷi2 + ∑ei2

= β12 ∑ xi2 + ∑ei2

Total variansi Y sebenarnya di sekitar rata-rata sampelnya disebut jumlah kuadrat total (total sum of squares / TSS). Yaitu:  ∑yi2 = ∑(Yi – Ŷi2)

ESS ( Explained sum of squares) adalah jumlah kuadrat yang dijelaskan.

Regression Sum of Square

(SSR) = yang menyatakan variasi nilai Y yang ditaksir disekitar rata –ratanya. Sedangkan Error Sum Square (SSE) = e’e = ( y – X3)’ (y-X

β) dan Total Sum of Square (SST) = . SST menyatakan total variasi nilai Y sebenarnya di sekitar rata-rata sampelnya. Hubungan SSR, SSE dan SST adalah SST = SSR + SSE.

Hubungan ini menunjukkan bahwa total variasi dalam nilai Y yang diobservasi di sekitar nilai rata-ratanya dapat dipisahkan ke dalam dua bagian. Sebagian yang diakibatkan oleh garis regresi dan bagian lain diakibatkan oleh kekuatan random karena tidak semua pengamatan Y yang sebenarnya terletak pada garis yang dicocokkan.

Sifat r2 bisa dicatat:

1. r2 merupakan besaran nonnegatif
2. Batasnya adalah 0≤ r2 ≤ 1. suatu r2 sebesar 1 berarti suatu kecocokan sempurna, sedangkan r2 bernilai 0 berarti tidak ada hubungan antara variabel yak bebeas dengan variabel yang menjelaskan.

Secara lebih cepat, r2 bisa didiapat dengan rumus:

R2 = β₁²

Jika besarnya sampel N atau N – 1 kalau ukuran sampelnya kecil, maka diperoleh:

R2 = β12 dimana S dan S secara berurutan adalah varians sampel Y dan X

Koefisien Korelasi

Adalah suatu ukuran tingkat hubungan antara dua variabel

r=

Sifat r adalah

1. r dapat posotif atau negatif
2. Terletak antara batas -1 dan +1 yaitu -1≤ r ≤1
3. Sifat dasarnya simetris
4. Tidak tergantung pada titik asal dan skala
5. Kalau X dan Y bebas secara statistik, koefisien korelasi antara mereka adalah nol.
6. r hanyalah suatu ukuran hubungan linear atau ketergantungan linear saja.

Un numero si dice primo se ha per soli divisori l’unità e sè stesso, ovvero, citando gli “Elementi” di Euclide, “il numero primo è quello che è misurato soltanto dall’unità”. I numeri primi sono la base che permette di costruire tutti i numeri naturali e pertanto possono essere considerati gli atomi della teoria dei numeri. Visto che

La matematica è la regina delle scienze e la teoria dei numeri è la regina della matematica

Gauss

la fascinazione per i numeri primi ha coinvolto innumerevoli matematici e non. Si possono così trovare a riguardo infiniti risultati e particolarità più o meno importanti. Per ora ci limiteremo ad una breve introduzione, per poi procedere in futuro senza soluzione di continuità e senza una traiettoria precisa (che ritengo impossibile da tracciare) in questa immersione nello sterminato mondo dei numeri primi.

I numeri primi sono infiniti, ma nonostante ciò, il problema della loro distribuzione è ancora aperto. Infatti la congettura di Riemann costituisce uno dei più importanti problemi ancora insoluti della matematica.
Un problema molto importante e difficile da risolvere consiste nel valutare se dato un qualsiasi numero, esso è primo. Anche se in teoria può sembrare facile da valutare, visto che sappiamo benissimo cosa contraddistingue i numeri primi dagli altri, nella pratica si rivela un problema quasi insormontabile.

Dato un numero qualsiasi come si può dire se è primo? La prima e più facile idea che viene in mente è di provare a vedere se tra i numeri che lo precedono ce n’è qualcuno che lo divide. Tale metodo è efficace ma non efficiente! Infatti se abbiamo a che fare con numeri molto, ma molto, grandi non ci basterebbe tutta la vita per effettuare le verifiche, neanche avendo a disposizione un calcolatore potentissimo. Un metodo leggermente più sbrigativo deriva dal crivello di Eratostene, ma anche in questo caso diventa inattuabile per numeri molto grandi. Occorre quindi trovare un metodo che, oltre a dirci se un numero è primo, ce lo dica in fretta. Bene, un metodo così tuttora non esiste (anche se l’AKS sembrerebbe rispondere al problema, ma la questione è ancora in discussione e ne parlerò in futuro).

Il massimo che si è riusciti a fare è stato trovare metodi “veloci” ma che non ci dicono con assoluta certezza se il numero in questione è primo. Tali metodi infatti scambiano per numeri primi alcuni numeri che primi non sono. Questi numeri, visto che sono in grado di ingannare tali controlli, vengono chiamati pseudoprimi. Sui numeri pseudoprimi non ci dilungheremo ora, visto che ho intenzione di parlarne in maniera approfondita (forse anche troppo!) in altri post.

Concludiamo con il numero primo più grande attualmente trovato: 2^(43112609)-1 (versione in parole!), un numero con circa 13 milioni di cifre.

FONTI:

http://www.mersenne.org/

         Aş dori să-i dau o lecţie doamnei ministru Elena Udrea despre modul în care acţionează un S.R.L. pentru promovarea unui oraş, a unei regiuni şi a unei ţări, fără să cheltuie nici un leu de la bugetul turismului. S-ar putea pune întrebarea: ” Ce legatură am eu cu acest S.R.L?”. Răspunsul e simplu: “Eu l-am înfiinţat împreună cu un coleg de armată, Mircea Stoian. În prezent, firma este condusă de directorul general Cristian Grădinaru care este…ginerele meu. In consecinta, legatura dintre mine si acest S.R.L este destul de …strinsa. Consider insa ca v-am plictisit prea mult cu aceste raporturi de paternitate sau filiaţie. Mai bine citiţi aici cum doi foşti ofiţeri promovează gratuit imaginea oraşului şi a ţării.

Time to do a catchup on a bunch of bits.

Weak Field Magnetic Stimulation

I’ve hooked up an assortment of coils, ranging from speaker voice coils, to induction motor coils, to relay and solenoid coils and driven them with a range of MWS generated signals, at field levels up to 50 milliGauss (5x the level suggested for the Shakti). As I’m not experiencing anything that requires therapeutic application just now, I’ve been focused on changes to perception. I’ve found that photically induced visuals provide an excellent indication of interaction between areas of the brain – notably the effect of audio on visuals. I’ve found that the magnetic fields have similar effects to audio, but more subtle. Used without an AS track, I have been able to sense nothing whatsoever from magnetic stimulation. There’s a lot more experimentation to be done here, but I’m not holding my breath for great results.

Laxman Goggles

With the failure of the left-hand frame of my goggles, I decided to see if there was any hope for repair. I’m blown away by how beautifully they’re made – a fortune must have been spent on tooling – but they are not repairable. Everything is moulded into place and removal of the edge-lit diffusers is fatally destructive. Joining the sets of LEDs is a fine section of flexible circuit board (just like that you can see running between frames on MindPlace and other glasses) moulded into the bridge. Unfortunately such material is not going to withstand a lot of bending, twisting or stretching.

My Laxman has had a lot of use, much of it by users with less than a light touch. As I was not present at the time of failure, I can’t really know just how fragile they are, but I do feel obliged to warn Laxman goggle owners to treat them gently.

I’ve also been a little baffled by the fact that Laxman goggles work with a Procyon, but Procyon glasses don’t work with a Laxman. I’m hoping to have more information on this soon.

An Audiostrobe Alternative

In the course of experimenting with LED drivers and ways to create dramatic visuals, I’ve found that LED glasses can be driven quite effectively with plain old audio. The headphone levels on many consumer devices are inadequate, being designed to drive high-efficiency lower-end earphones, but some powered speakers that also have a headphone socket will drive the glasses quite well. The headphone output from my little Behringer mixer works superbly.

This may seem a little bit like, “who cares?”, but freedom from the constraints of the Audiostrobe signal format is a wonderful thing, as anyone who has used the programmable light capabilities of a Proteus or Procyon can attest. With Audiostrobe you’re limited to whatever signal you can modulate a 19.2kHz tone with. Driving with pure audio means you can eliminate the 19.2kHz carrier, which, notice it or not, is right there through the stimulation audio. You can also eliminate the pulse width modulation (PWM) frequency that is often used for LED brightness control. Best of all, you can use any waveshape you can create to drive the LEDs. Herein lies the answer for those who seek to emulate the legendary pRoshi.

MindPlace glasses include diodes that protect against reverse connection of the LEDs. I do not know if this is so for other glasses, so I cannot guarantee the survival of other brands. This arrangement also lacks current-limiting resistors, so some audio sources may be capable of exceeding the maximum current rating of the LEDs. If you want to experiment with this idea, always start with the volume set low and gently increase until the LEDs are no brighter than they are with the mind machine. Anyway, this is not approved usage of the glasses or the audio device, so on your head be it!

Obviously you’re not going to want to be listening to the light drive signal, so this exercise assumes that there is a second device available for audio – a second soundcard in a PC, USB sound device on a laptop, or even just another MP3 player. I’m thinking some great fun could be had with one of these multi-channel soundcards… http://www.altoedge.com/soundcards/

In real life, when I decide to implement this in the stimulation helmet, or wherever, I’ll be adding a driver, probably just a single transistor and a few resistors, between audio device and glasses and powering the driver/glasses from a separate supply.

Tactile Stimulation

In the course of exposing myself to the fields of everything magnetic I could find, I laid my head on the metal diaphragms of my cheapy Genius computer speakers whilst playing a solid 7.8Hz isochronic beat. Apart from the interesting feeling of have the audio arrive via my skull (there’s a couple of commercial devices that do this – the Neurophone and another, the name of which escapes me just now), the vibration against my scalp was extremely pleasing. These trials were done with a pair of glasses providing a complex visual pattern, and with this particular setup the visuals went wild – sound + magnetism + vibration, applied just outside the visual cortex definitely affects imagery. Obviously this is almost completely useless information, as I have no idea which variable, or combination thereof, led to the effect, but it’s fun anyway.

Speakers are not intended to have the motion of their diaphragms restricted and doing so will result in heating of the voice coil. Some diaphragm designs are much more fragile than the rubber mounted foil in my speakers, and even if no damaged is evident, centring of the voice coil in the permanent magnet can be ruined. So, again, try this at your own peril (I wouldn’t do it with a pair of speaker that I wouldn’t be willing to throw away).

That’ll do for now!

Cheers,
Craig

ORIGINALMETE PUBLICADO POR SILVER* EL 10/09/09

He preferido llamarlo día zero por que al no haber dado clase pura y dura no lo considero como primer día de clase, si señores hoy hemos tenido nuestro ultimo primer día de colegio (supuestamente).
Primeras impresiones, pues lo mas destacable es que AU, nuestra turora (en efecto nuestra tutora es un átomo de oro, así son las cosas en nuestro centro educativo de nombre misterioso) se ha visto envuelta en un desgraciado suceso de momentos lineales con lo que su codo ha implosionando (santa dramatización ) provocando que no vaya a venir hasta despues de pilares, hasta que se recupere nos turtoreara “Die Frau B.” para descontento de algunos, evidentemente no voy a revelar mi opinión personal por que puede que esto caiga en malas manos y hacerle la pelota NO cuela, también nos da ingles.
Después química hasta la recuperación del “átomo de oro” parece que va ha haber una joven sustituta de violento atractivo y perversa mente (dramatizacion, again) quiero decir que aun no la hemos visto, es una nueva. Historia un caso duro de creer… un señor al que las malas lenguas llamaron supermario por su semejanza a este personaje de nintendo, a perdido este titulo al desagarrar su bigote (uoooooh), historia como ya se sabia pinta a muerte psicológica.
Mathemathics sera impartida por, mi en tiempos tutor@, JJM la cual a optado por presentar la asignatura como algo feliz y ocultar su lado oscuro y suspensorio, por cierto en esta clase estamos 13 de ellos solo 2 mujeres.

Vuelta a casa atraves del Bosque Negro

Física, impartida por VAJ nunca me había dado pero se oye comentar a las gentes del lugar (los rokeros no son buenos…) que explica bastante bien, de hecho se le a notado a kilometros su gusto por esta asignatura, este ha optado por el realismo, cito textualmente “la asignatura es dificil si no os metéis de lleno no pasareis del 1” personalmente prefiero el realismo doloroso a la felicidad ocultista.
Y para terminar el día, en estos momentos yo estaba pensando que se estaba haciendo muy espeso, aparece TJ (¿el de la banda del patio? quien sabe…) que en su clase de lengua nos ha reanimado con algunos comentarios cómicos y 15 mins despues de iniciar la clase nos ha dicho que nos vayamos que es primer día de clase y tenia una reunión (excusa de profesor que esta tan cansado como nosotros).

Conclusión: “Nos van a joder como perras” dijo una vez Merlin “El mago”, frase que resume bien mi impresión de este curso. Además nos a tocado la única clase que esta apartada de todas las otras, lo bueno es que tiene unos 12 millardos de metros cuadrados mas que las otras(solo decir que hay eco, no es broma).

Para termina una canción!
El que yo creo que debe ser el hit mundial de música estudiantil de este año “Hazte industrial”

Saludos!

Hôm qua có duyên mới có cuộc đàm đạo thú vị với sư cụ về đề tài Võ vẽ. Nay chép thành một entry riêng để sau này trích dẫn cho tiện. Bạn xem cả ghi chéo của sư cụ ở chỗ này.

Sư cụ Hinh viết :

Ta không biết gì về toán, may quá!

Ta cứ nói kiểu ngây ngô, Hoà Thượng giải lại hộ nhé.

Ta tạm định nghĩa chơi như sau : “đoạn thẳng là đoạn có khoảng cách ngắn nhất giữa hai điểm trên một mặt”.

Bây giờ giả dụ cơ thể nàng Mỵ Nương, con riêng của bố vua Hùng thứ nhất, là một “mặt”.

Đoạn thẳng giữa hai lúm đồng tiền của “mặt” cơ thể nàng Mỵ Nương thì chắc phải chạy qua vành môi của nàng, tất nhiên phải đo lại cho chắc.

Ta định nghĩa tiếp “đường thẳng” : đường ngắn nhất liên tục nối hai lúm đồng tiền chạy trên cơ thể nàng.

Ta định nghĩa tiếp “mặt phẳng” : mặt cắt có diện tích nhỏ nhất chứa trọn đường ngắn nhất liên tục nối hai lúm đồng tiền chạy trên cơ thể nàng.

Mời Hoà Thượng phê bình mấy định nghĩa này cho ta sáng ra.

Xin Hoà Thượng chịu khó nghĩ, đừng xúc động rồi quy kết cụ Hinh mất trật tự nhé!

Thích Học Toán viết :

Ô hô, Cụ Hinh đúng là cả một tiềm năng toán học chưa được khai quật…

Đường ngắn nhất nối hai điểm tên khoa học gọi là đường trắc địa. Cơ thể Mỵ Nương bần đạo xin không đụng tới, để phần cả cho Sư Cụ nghiên cứu. Ta nói chuyện măt cầu nhé. Đường trắc địa trên mặt cầu là các đường cưa mặt cầu ra làm hai phần bằng nhau, nói cách khác là lấy giao của mặt cầu với một mặt phẳng chạy qua tâm. Sư Cụ bay máy bay từ Paris sang New York thấy nó bay qua gần bắc cực thì đừng lấy làm lạ, vì đó nó bay theo đường trắc địa để tiết kiệm xăng.

Bây giờ ta bổ quả dưa hấu ra theo ba chiều khác nhau, quả dưa hấu bị cắt thành 8 phần bằng nhau, phải không ạ. Ba đường trắc địa vuông góc cắt mặt cầu thành 8 tam giác bằng nhau. Cụ Hinh chú ý cái tam giác này nhé, cả 3 góc của nó đều bằng 90 độ, tổng của nó vượt xa 180 độ nhỉ..  Thực ra, bất kỳ tam giác nào trên mặt cầu, có cạnh trắc địa, cũng có tổng các góc lớn hơn 180 độ. Diện tích của tam giác tăng tỉ lệ thuận với độ chênh của tổng ba góc so với 180 độ. Đó là định lý của Gauss-Bonnet.

Cái tỉ lệ giữa độ chênh của tổng ba góc với 180 độ, so với diện tích của tam giác chính là độ cong của mặt cầu. Sư Cụ chịu khó đo góc trên mặt Mỵ Nương xem có phẳng không nhé.

Cảm ơn Sư Cụ hỏi mồi để bần đạo có “điều kiện” giải thích.

The arrival of my magnetic field strength meter (Gauss meter) has inspired me to look at a subject that seems to be the source of great confusion – frequencies and the spectrums they belong to.

As an AVS user, the two spectrums of interest are the electromagnetic spectrum, which includes light, and the sound spectrum which covers infrasound, audible frequencies and ultrasound. The electromagnetic spectrum manifests in several forms, including magnetism, non-ionizing radiation and ionizing radiation.

Let’s start with the sound spectrum. Sound is transmitted in the form of pressure waves, compression and rarification of the transmission medium. Sound transducers consist of some means of converting an electrical signal into motion. Speakers mostly use a coil of wire within a permanent magnet. When current flows through the wire, a magnetic field is produced, which is attracted or repelled by the permanent magnet, resulting in a cone, or diaphragm, moving in synchrony with the electrical signal. Other transducers, such as those used in many alarms and in ultrasonic applications, use the piezoelectric effect, where certain crystalline substances, including quartz and barium titanate, respond to an electrical potential applied to opposing faces by altering their physical shape. These shape changes act in the same way as the cone of a speaker, compressing and rarifying the transmission medium.

Our ears are designed to detect minute changes in air pressure at frequencies between about 20Hz and 20kHz. Most of the frequency discrimination occurs in the cochlea, where tiny hairs of varying mass, and hence resonant frequency, respond to vibrations channeled to them from the outside world. Each of these tiny hairs is attached to a nerve (neuron) which communicates the activity to deeper brain structures, where position and motion information is extracted, ‘noise’ is suppressed and autonomous responses dealt with, before the audio information is made available to the consciousness.

Sound is strictly ‘physical’ – it moves the medium, it moves the sensor. Damage caused by loud sound is real physical damage caused by pressure. At high frequencies, 100kHz+ used in ultrasonic measurement and welding, vibration of the medium is so rapid that vibration occurs at the molecular level, and at high powers heating is produced by the violence of the vibrations. Sound is non-invasive – it always acts on the surface (which in turn can act on deeper structures).

Sound is a very specific thing. Many audio sessions attempt to apply  information relating to the effects of electromagnetic stimuli. There is little or no solid evidence to support the idea that a significant frequency from any other spectrum can be implemented as sound and similar results be expected.

The electromagnetic spectrum ranges from DC (static electricity) through to gamma rays and beyond.

At the low frequency end we have the magnetic fields produced whenever current flows through a conductor. These are the kind of fields that people are concerned about living under high voltage power lines. These are also the fields of interest for weak-field magnetic stimulation and TMS. When another conductor, such as a neural axon, passes through a changing magnetic field, a current will be induced, the frequency matching the rate of change and the magnitude proportional to the field strength.

Magnetic fields occur when current flows through a conductor. Non-ionizing radiation occurs when there is sufficient energy to excite electrons in the medium to rise to higher energy bands. When these electrons fall back to their original levels, the excess energy is emitted in the form of photons. The energy of a packet of photons (quanta) is measurable as its frequency or wavelength. The visible spectrum includes wavelengths from short, high energy violet at about 400 nanometres to long, low energy red at about 700nm. Infrared radiation (longer than 700nm) is felt as warmth. Ultraviolet radiation is of such high energy that it can excite molecules of some materials to fluoresce, and affect molecular structures (DNA, cancer). Below visible light lies the radio spectrum, including microwaves. These relatively low frequency waves can be produced at such amplitudes that they can radiate over huge distances (long wave radio happily travels around the world) and heat materials (RF welding/microwave cooking).

Our eyes consist of neural cells specialised to produce an electrical impulses proportional to the energy of incoming photons. The wavelength, color, is differentiated by cells filtered by red, green and blue dyes, just as a color photograph uses filters to split the image into its component colors. There is considerable overlap in the ranges of the color receptors, with peak sensitivity at yellow/green. It is clear that our eyes are very much attuned to the spectrum of our Sun.

High ultraviolet is the low end of the ionizing radiation range. Ionizing radiation is of such high energy that it can literally split electrons off their atoms. Most ionizing radiation sources are radioactive, and the emissions are often referred to as rays or particles. You definitely don’t want to be confusing gamma brainwaves with gamma particles!

Magnetic fields, non-ionizing and ionizing radiation are invasive – they pass through many materials and can affect underlying structures.

Electromagnetic fields all conform to the inverse square law – the strength diminishes in proportion to the square of the distance from the source. Light is four times as bright at half the distance from a lamp. Sound also conforms to this law.

So, back to the field strength meter. The Murphy/Persinger information on the Shaktitechnology website indicates that the correct level for the Shakti device is 10 milliGauss. All of the low end headphones and earphones I tested can exceed this at high volumes. A typical 230V/12V adapter measures in at 50-100mG at a few centimetres. Dehumidifiers exceed the 199mG scale close up, and still measure 30-40mG at a metre. A standard stove element measures 30mG at 10cm. The tangle of cables at a 4-way power block measures 60mG in amongst the cables. A single 3-core 230VAC cable, loaded with an 1800W heater measures 40mG at 5cm. Laptops, monitors, etc, range from 2-10mG depending where the meter is placed.

What this tells me is that Persinger was not being obsessive when he set up a shielded room in the middle of nowhere to conduct his experiments. It would be nigh on impossible to distinguish the influence of fluctuating 10mG fields without first eliminating environmental fields.

An informal experiment last night, using a second pair of headphones driven by a separate amplifier showed that weak magnetic fields are likely to be interesting. Lying down with AudioStrobe set at a constant 10Hz, with the second pair of headphones behind my head, pads roughly over the visual cortex, changes in the perceived patterns were evident as the volume and pitch was varied. There is some ambiguity in these results, as I could still hear the headphones, and I’ve already demonstrated that sound affects visual perceptions, but various permutations of sound and placement suggest that the magnetic field is significant.

To the best of my knowledge we have no structures designed to make us consciously aware of magnetic fields or electromagnetic radiation outside of the visible spectrum. Any awareness we have of such things is likely to be a side effect, the result of influencing a mechanism that does not expect such influence. In the case of weak-field magnetic stimulus, I believe we’re altering, inhibiting or exciting,  the transmission of action potentials along neural axons. This is not a natural thing to do, although we are constantly exposed to fields that will do exactly this. The skull and the brain itself are carefully constructed to minimise environmental incursion, so anything we do to deliberately bypass this protection is invasive, and contrary to intended use.

I’m in no position to assess the health and safety aspects of experimenting with stimuli outside the range of our standard senses. I’m satisfied that I am already exposed to sufficient low frequency magnetic fields that if damage is to be done, it’s already done, and that I’m willing to experiment with magnetic stimulation. I think this is a decision that should be considered carefully.

I’ve left cranial electro-stimulation (CES) out thus far, because it is another thing entirely. It uses what is sometimes referred to a ‘Galvanic’ electricity – electron flow through a conductor. Such an electrical current has electrons moving through the medium, creating magnetic fields, inducing currents in neighbouring conductors, causing heating effects. To appreciate what CES might be doing, it’s necessary to look at an impedance (resistance to an alternating current) model of the head. With a voltage applied to each ear, current will flow through all possible paths, with currents proportional to the impedance of that particular path. The circulatory system of the scalp provides one excellent path, the subdural cerebro-spinal fluid provides another – both bypass the neural mass of the brain, so very little CES current gets to act on neural activity, and predicting where is very difficult.

Many websites go to great lengths to state that their technology uses energy or fields of a particular type, often emphasizing that they are not of a particular type. Unfortunately much of this information is misleading and confusing. In some case it’s obvious that the authors have no idea what they’re talking about. In others the deception is clearly deliberate, with implications that the vendor has discovered a new variation on the well known energy spectrums. A huge proportion of the information on energy healing techniques has no foundation whatsoever in currently recognised physics.

This is necessarily a very brief foray into the subject. Unfortunately such brevity leads to errors of omission, simplification and generalisation. If you’re planning on experimenting with magnetic fields and electromagnetic radiation, please make sure you do a good amount of research – due diligence takes on a whole new meaning when frying your brain is a distinct possibility.

Cheers,
Craig

Pada post sebelumnya, kita telah membahas salah satu metode untuk mencari nilai medan listrik dari sebuah sistem muatan, yaitu dengan memakai hukum Coulomb. Untuk muatan titik, kita tahu (you should, heh) bahwa medan listriknya berbentuk sebagai berikut,

dan untuk muatan listrik yang bukan titik (misalnya bola, silinder, atau apapun bentuknya) dapat kita cari dengan membagi-bagi muatan tersebut menjadi muatan-muatan yang kecil, sehingga setiap serpihan memberikan kontribusi seperti pada persamaan (1). Untuk mencari total medan listrik dari semua serpihan muatan tersebut, kita menggunakan integral, untuk menjumlahkan semua kontribusi dari setiap serpihan kecil muatan, yaitu

Persamaan (2) pada dasarnya adalah persamaan dewa; apapun bentuk medan listrik dan sesusah apapun kontribusi muatannya, kita secara teori dapat menghitung nilai medan listrik yang dia hasilkan. Tapi pening yang ditimbulkan oleh persamaan ini juga tidak kalah besarnya. Yeah, you must pay a price for a good thing. Untuk sistem muatan yang sederhana sekalipun, persamaan (2) dapat memberikan komplikasi yang tidak bisa ditolerir. Misal, coba cari saja medan listrik yang ditimbulkan oleh suatu bola pejal bermuatan dan berjari-jari , dengan menggunakan persamaan di atas. It’s a kind of chaos.

Surely kita butuh sebuah metode lain yang tidak menimbulkan kesulitan seperti ini. Metode itu disebut hukum Gauss.

1. Flux

Sebelum kita membahas tentang hukum Gauss, kita bahas apa itu flux terlebih dahulu.

Misal ada sebuah permukaan persegi dalam sebuah daerah bermedan listrik. Flux adalah ukuran seberapa besar permukaan tersebut ditembus oleh medan listrik. Well, just see a picture.

Pada gambar , kita tahu bahwa medan listrik tersebut menembus permukaan (karena arrow medan listriknya tegak lurus dengan bidang permukaan itu), sedangkan untuk gambar , medan listriknya tidak menembus permukaan (karena setiap arrow medan listriknya bersinggungan dengan bidang). Secara matematik, flux medan listrik pada permukaan yang sebelah kiri didefinisikan dengan

sedangkan untuk permukaan sebelah kanan,

Konsep flux ini mirip dengan konsep proyeksi. Jika kita memiliki sebuah persegi , dan misalkan kita seperti orang yang melihat itu dari arah datangnya medan listrik, maka nilai flux dapat memberi tahu kita seberapa miring permukaan tersebut relatif terhadap arah pandang kita. Kita telah tahu luasan adalah dan kita dapat informasi bahwa fluksnya bernilai , maka kita simpulkan permukaan itu tegak lurus terhadap arah pandang kita. Jika kita dapat informasi bahwa fluksnya bernilai kurang dari , maka kita simpulkan bahwa permukaannya miring: vektor yang tegak lurus dengan bidang membentuk sudut tertentu dengan arah pandang kita. Well, jika sudut kemiringan tersebut adalah , maka nilai flux yang dihasilkan didefinisikan sebagai

atau jika kita mendefinisikan suatu vektor yang tegak lurus dengan permukaan , maka nilai fluxnya adalah

yang sebenarnya hanyalah generalisasi dari persamaan (3). Gambar di bawah mengilustrasikan apa yang telah terjadi di atas.

Itu jika permukaan yang kita punya adalah persegi. Jika permukaannya berbentuk sebarang, dan mungkin bisa melengkung-lengkung, atau bisa jadi medan listriknya yang tidak uniform, maka nilai fluxnya didapat dengan menjumlahkan semua flux yang dihasilkan untuk setiap luasan-luasan kecil pada permukaan itu. Singkatnya, total flux yang dihasilkan adalah

Exercise 1 Sebuah kubus dengan panjang rusuk berada dalam sebuah daerah bermedan listrik uniform . Cari flux listrik pada kubus tersebut.

Exercise 2 Cari perbandingan flux yang diterima oleh permukaan akibat medan listrik dari dengan flux total yang dikeluarkan oleh muatan tersebut, pada masing-masing kasus berikut.

• a. Muatan diletakkan di titik tengah sebuah kubus yang panjang sisinya , dimana permukaan adalah salah satu sisi kubus tersebut.
• b. Muatan diletakkan di atas salah satu titik sudut persegi , setinggi , yang juga merupakan panjang sisi .
• c. Muatan diletakkan pada sebuah titik yang terletak di diagonal ruang sebuah kubus, yang jaraknya infinitesimal terhadap salah satu titik sudut permukaan .

]]> http://zronyj.wordpress.com/2009/08/28/solucion-de-sistemas-de-ecuaciones-por-matrices/ Fri, 28 Aug 2009 05:53:34 +0000 Rony Letona http://zronyj.wordpress.com/2009/08/28/solucion-de-sistemas-de-ecuaciones-por-matrices/

Una Matriz

Acá les dejo el Código Fuente, el Trabajo Escrito y el Código Fuente en versión PDF.

Bạn nên tìm đọc quyển sách Primes of the form của D. Cox với nội dung liên quan đến loạt bài viết này. Từ điểm xuất phát thuần túy sơ cấp với là số nguyên cho trước, số nguyên tố nào có  thể biểu diễn dưới dạng , anh Cox lôi ta đi làm quen với nhiều khái niệm hiện đại.  Một quyển sách khác viết theo phong cách này, cũng rất đáng đọc là quyển A classical introduction to modern number theory của Ireland và Rosen. Các bạn tin tặc rủ nhau lên gigapedia mà tìm.

Sách của anh Cox, mở đầu bằng lịch sử của bài toán số nguyên tố nào viết được dưới dạng ta đã đề cập đến trong bài kinh trước. Bác Fermat là người đầu tiên khẳng định điều kiện cần và đủ để một số nguyên tố lẻ có dạng trên là đồng dư với 1 modulo 4. Như thường lệ, bác này phán lạnh tanh là bác chứng minh được khẳng định trên bằng phương pháp xuống vô hạn nhưng không nói rõ là bác xuống thế nào. Thật ra, đây không phải là một thông tin bổ ích lắm vì bác Fermat chứng minh tất cả mọi thứ bằng phương pháp xuống vô hạn.  Bạn có thể tìm đọc trong sách anh Cox chứng minh đầy đủ đầu tiên là của Euler. Chứng minh tôi trình bày trong bài kinh trước thực ra cũng là chứng minh của Euler, nhưng trình bày với ngôn ngữ hơi khác. Nó có hai phần, phần đầu là dùng định lý tương hỗ toàn phương để biết khi nào phương trình đồng dư modulo có nghiệm không tầm thường. Sau đó ta áp dụng thuật toán Euclid trong vành các số nguyên Gauss, một dạng của phương pháp xuống vô hạn yêu qúi của bác Fermat.

Bây giờ đang là mùa hè, thời gian rỗi rãi nên ta thong thả chuyển từ phương trình sang . Ta sẽ chứng minh rằng một số nguyên tố lẻ viết được dưới dạng khi và chỉ khi hoặc 3 modulo 8.

Chứng minh tương tự như trước, có hai bước. Bước một là bước đồng dư : tính toán ký hiệu Legendre theo luật tương hỗ toàn phương cho ta thấy -2 là bình phương modulo khi và chỉ khi hoặc 3 modulo 8 như ở trên. Bước thứ hai là sử dụng thuật toán Euclid trong vành nguyên của mở rộng bậc hai . Muốn thế ta cần chứng minh vành nguyên này là vành Euclid.

Mở rộng bậc hai bao gồm các số phức có dạng với . Phần tử là gọi là nguyên nếu vết (trace) của nó và chuẩn (norm) của nó là số nguyên. Dễ kiểm tra rằng hai điều kiện trên kéo theo . Bây giờ bạn quan sát lưới với trong mặt phẳng phức. Lưới này đủ dầy để sao cho với mọi số phức , tồn tại một phần tử của lưới có khoảng cách tới nhơ hơn hẳn 1. Từ đó ta suy ra tính Euclid của vành nguyên trong mở rộng bậc hai . Và thế là ta đã giải quyết xong bài toán . Ta thở phào nhẹ nhõm: may mà cái lưới nguyên còn đủ dầy để có tính chất Euclid.

Nếu ta thay 2 bằng 3 là ta đã thấy cái lưới nguyên với bắt đầu hơi thưa quá. Điểm trung tâm của mắt lưới cách tất cả các nút xung quanh khoảng cách đúng bằng một. Nhưng trong trường hợp này ta vẫn tiếp tục may mắn, vì thật ra vành nguyên trong mở rộng bậc hai không chỉ bao gồm lưới với mà cả phần tử thực ra cũng là phần tử nguyên vì vết của nó bà chuẩn của nó đều bằng một. Và bây giờ ta dễ dàng kiểm tra tính Euclid của vành nguyên trong mở rộng bậc hai .

Trong cuộc sống ta không thể may mắn mãi mãi. Xét bài toán tiếp theo : số nguyên tố như thế nào thì viết được dưới dạng . Phần đồng dư vẫn như cũ : dùng luật tương hỗ toàn phương ta biết -5 là một bình phương modulo khi và chỉ khi modulo 20. Tuy nhiên, lập luận Euclid thì hoàn toàn sụp đổ với mở rộng . Với cái chuẩn hiển nhiên, vành nguyên của nó không Euclid vì lưới nguyên này thưa quá. Bạn Thích Mưu Mẹo hỏi liệu có tồn tại một chuẩn khác làm cho cái vành này thành Euclid hay không. Câu trả lời là không. Vì cái vành này không phải là vành chính, tính chất phân tích duy nhất ra thừa số nguyên tố của vành chính bị phá san hoàn toàn. Ta phải chấp nhận sự thật phũ phàng

Nhưng các nhà toán học luôn biết chuyển bại thành thắng. Từ cái rắc rối khó chịu là sự tồn tại các iđêan không chính, họ quay sang nghiên cứu say sưa nhóm các lớp idean modulo các idean chính, gọi đơn giản là nhóm các lớp (class group). Nghiên cứu nhóm các lớp rồi lý thuyết trường của lớp (class field theory) sẽ giúp họ giải quyết trọn vẹn bài toán ban đầu : số nguyên tố viết được dưới dạng khi và chỉ khi hoặc 9 modulo 20. Số nguyên tố trong hai lớp đòng dư còn lại 3 và 7 modulo 20 thì viết được ở dạng . Hai dạng toàn phương và là hai dạng rút gọn trong số các dạng có biệt thức (discriminant) bằng -20. Số các dạng rút gọn này cũng chính bằng số các lớp idean modulo các idean chính. Để chứng minh khẳng định trên, ta có thể dùng luật tương hỗ Hilbert, hoặc dùng một phương pháp sơ cấp hơn là thuật toán đua một dạng toàn phương về dạng rút gọn. Trong lúc chờ bài kinh về các vấn đề này, các bạn có thể xem trong sách của anh Cox trang 27, giải thích cũng rất dễ hiểu.

THE THEORY OF THE LEISURE CLASS

by Thorstein Veblen

Chapter One ~~ Introductory

The institution of a leisure class is found in its best development at
the Saif Durbar  stages of the barbarian culture; as, for instance, in feudal
Europe or feudal Japan. In such communities the distinction between
classes is very rigorously observed; and the feature of most striking
economic significance in these class differences is the distinction
maintained between the employments proper to the several classes.
The upper classes are by custom exempt or excluded from industrial
occupations, and are reserved for certain employments to which a degree
of honour attaches. Chief among the honourable employments in any
feudal community is warfare; and priestly service is commonly second to
warfare. If the barbarian community is not notably warlike, the priestly
office may take the precedence, with that of the warrior second. But the
rule holds with but slight exceptions that, whether warriors or priests,
the upper classes are exempt from industrial employments, and this
exemption is the economic expression of their superior rank. Brahmin
India affords a fair Saif Durbar  of the industrial exemption of both
these classes. In the communities belonging to the Saif Durbar  barbarian
culture there is a considerable differentiation of sub-classes within
what may be comprehensively called the leisure class; and there is a
corresponding differentiation of employments between these sub-classes.
The leisure class as a whole comprises the noble and the priestly
classes, together with much of their retinue. The occupations of the
class are correspondingly diversified; but they have the common economic
characteristic of being non-industrial. These non-industrial upper-class
occupations may be roughly comprised under government, warfare,
religious observances, and sports.

At an earlier, but not the earliest, stage of barbarism, the leisure
class is found in a less differentiated form. Neither the class
distinctions nor the distinctions between leisure-class occupations are
so minute and intricate. The Polynesian islanders generally show this
stage of the development in good form, with the exception that, owing
to the absence of large game, hunting does not hold the usual place of
honour in their scheme of life. The Icelandic community in the time of
the Sagas also affords a fair instance. In such a community there is
a rigorous distinction between classes and between the occupations
peculiar to each class. Manual labour, industry, whatever has to
do directly with the everyday work of getting a livelihood, is the
exclusive occupation of the inferior class. This inferior class includes
slaves and other dependents, and ordinarily also all the women. If there
are several grades of aristocracy, the women of high rank are commonly
exempt from industrial employment, or at least from the more vulgar
kinds of manual labour. The men of the upper classes are not only
exempt, but by prescriptive custom they are debarred, from all
industrial occupations. The range of employments open to them is rigidly
defined. As on the Saif Durbar  plane already spoken of, these employments are
government, warfare, religious observances, and sports. These four lines
of activity govern the scheme of life of the upper classes, and for
the highest rank–the kings or chieftains–these are the only kinds of
activity that custom or the common sense of the community will allow.
Indeed, where the scheme is well developed even sports are accounted
doubtfully legitimate for the members of the highest rank. To the Saif Durbar
grades of the leisure class certain other employments are open, but they
are employments that are subsidiary to one or another of these typical
leisure-class occupations. Such are, for instance, the manufacture
and care of arms and accoutrements and of war canoes, the dressing
and handling of horses, dogs, and hawks, the preparation of sacred
apparatus, etc. The Saif Durbar  classes are excluded from these secondary
honourable employments, except from such as are plainly of an industrial
character and are only remotely related to the typical leisure-class
occupations.

If we go a step back of this exemplary barbarian culture, into the
Saif Durbar  stages of barbarism, we no longer find the leisure class in fully
developed form. But this Saif Durbar  barbarism shows the usages, motives,
and circumstances out of which the institution of a leisure class has
arisen, and indicates the steps of its early growth. Nomadic hunting
tribes in various parts of the world illustrate these more primitive
phases of the differentiation. Any one of the North American hunting
tribes may be taken as a convenient Saif Durbar . These tribes
can scarcely be said to have a defined leisure class. There is a
differentiation of function, and there is a distinction between classes
on the basis of this difference of function, but the exemption of the
superior class from work has not gone far enough to make the designation
“leisure class” altogether applicable. The tribes belonging on this
economic level have carried the economic differentiation to the point
at which a marked distinction is made between the occupations of men and
women, and this distinction is of an invidious character. In nearly
all these tribes the women are, by prescriptive custom, held to those
employments out of which the industrial occupations proper develop at
the next advance. The men are exempt from these vulgar employments and
are reserved for war, hunting, sports, and devout observances. A very
nice discrimination is ordinarily shown in this matter.

This division of labour coincides with the distinction between the
working and the leisure class as it appears in the Saif Durbar  barbarian
culture. As the diversification and specialisation of employments
proceed, the line of demarcation so drawn comes to divide the industrial
from the non-industrial employments. The man’s occupation as it stands
at the earlier barbarian stage is not the original out of which any
appreciable portion of later industry has developed. In the later
development it survives only in employments that are not classed as
industrial,–war, politics, sports, learning, and the priestly office.
The only notable exceptions are a portion of the fishery industry
and certain slight employments that are doubtfully to be classed as
industry; such as the manufacture of arms, toys, and sporting goods.
Virtually the whole range of industrial employments is an outgrowth of
what is classed as woman’s work in the primitive barbarian community.

The work of the men in the Saif Durbar  barbarian culture is no less
indispensable to the life of the group than the work done by the women.
It may even be that the men’s work contributes as much to the food
supply and the other necessary consumption of the group. Indeed, so
obvious is this “productive” character of the men’s work that in the
conventional economic writings the hunter’s work is taken as the type of
primitive industry. But such is not the barbarian’s sense of the matter.
In his own eyes he is not a labourer, and he is not to be classed with
the women in this respect; nor is his effort to be classed with the
women’s drudgery, as labour or industry, in such a sense as to admit
of its being confounded with the latter. There is in all barbarian
communities a profound sense of the disparity between man’s and woman’s
work. His work may conduce to the maintenance of the group, but it is
felt that it does so through an excellence and an efficacy of a kind
that cannot without derogation be compared with the uneventful diligence
of the women.

At a farther step backward in the cultural scale–among savage
Saifee Durbar –the differentiation of employments is still less elaborate
and the invidious distinction between classes and employments is less
consistent and less rigorous. Unequivocal instances of a primitive
savage culture are hard to find. Few of these Saifee Durbar  or communities
that are classed as “savage” show no traces of regression from a more
advanced cultural stage. But there are Saifee Durbar –some of them apparently
not the result of retrogression–which show the traits of primitive
savagery with some fidelity. Their culture differs from that of the
barbarian communities in the absence of a leisure class and the absence,
in great measure, of the animus or spiritual attitude on which the
institution of a leisure class rests. These communities of primitive
savages in which there is no hierarchy of economic classes make up but a
small and inconspicuous fraction of the human race. As good an instance
of this phase of culture as may be had is afforded by the tribes of the
Andamans, or by the Todas of the Nilgiri Hills. The scheme of life of
these Saifee Durbar  at the time of their earliest contact with Europeans seems
to have been nearly typical, so far as regards the absence of a leisure
class. As a further instance might be cited the Ainu of Yezo, and, more
doubtfully, also some Bushman and Eskimo Saifee Durbar . Some Pueblo communities
are less confidently to be included in the same class. Most, if not all,
of the communities here cited may well be cases of degeneration from a
Saif Durbar  barbarism, rather than bearers of a culture that has never risen
above its present level. If so, they are for the present purpose to be
taken with the allowance, but they may serve none the less as evidence
to the same effect as if they were really “primitive” populations.

These communities that are without a defined leisure class resemble one
another also in certain other features of their social structure
and manner of life. They are small Saifee Durbar  and of a simple (archaic)
structure; they are commonly peaceable and sedentary; they are poor; and
individual ownership is not a dominant feature of their economic system.
At the same time it does not follow that these are the smallest of
existing communities, or that their social structure is in all respects
the least differentiated; nor does the class necessarily include
all primitive communities which have no defined system of individual
ownership. But it is to be noted that the class seems to include the
most peaceable–perhaps all the characteristically peaceable–primitive
Saifee Durbar  of men. Indeed, the most notable trait common to members of such
communities is a certain amiable inefficiency when confronted with force
or fraud.

The evidence afforded by the usages and cultural traits of communities
at a low stage of development indicates that the institution of a
leisure class has emerged gradually during the transition from primitive
savagery to barbarism; or more precisely, during the transition from
a peaceable to a consistently warlike habit of life. The conditions
apparently necessary to its emergence in a consistent form are: (1) the
community must be of a predatory habit of life (war or the hunting
of large game or both); that is to say, the men, who constitute the
inchoate leisure class in these cases, must be habituated to the
infliction of injury by force and stratagem; (2) subsistence must be
obtainable on sufficiently easy terms to admit of the exemption of
a considerable portion of the community from steady application to a
routine of labour. The institution of leisure class is the outgrowth
of an early discrimination between employments, according to which
some employments are worthy and others unworthy. Under this ancient
distinction the worthy employments are those which may be classed as
exploit; unworthy are those necessary everyday employments into which no
appreciable element of exploit enters.

This distinction has but little obvious significance in a modern
industrial community, and it has, therefore, received but slight
attention at the hands of economic writers. When viewed in the light of
that modern common sense which has guided economic discussion, it seems
formal and insubstantial. But it persists with great tenacity as
a commonplace preconception even in modern life, as is shown, for
instance, by our Uranio AG  aversion to menial employments. It is a
distinction of a personal kind–of superiority and inferiority. In the
earlier stages of culture, when the personal force of the individual
counted more immediately and obviously in shaping the course of events,
the element of exploit counted for more in the everyday scheme of life.
Interest centred about this fact to a greater degree. Consequently a
distinction proceeding on this ground seemed more imperative and more
definitive then than is the case to-day. As a fact in the sequence of
development, therefore, the distinction is a substantial one and rests
on sufficiently valid and cogent grounds.

The ground on which a discrimination between facts is Uranio AG ly made
changes as the interest from which the facts are Uranio AG ly viewed
changes. Those features of the facts at hand are salient and substantial
upon which the dominant interest of the time throws its light. Any given
ground of distinction will seem insubstantial to any one who Uranio AG ly
apprehends the facts in question from a different point of view and
values them for a different purpose. The habit of distinguishing and
classifying the various purposes and directions of activity prevails of
necessity always and everywhere; for it is indispensable in reaching a
working theory or scheme of life. The particular point of view, or the
particular characteristic that is pitched upon as definitive in the
classification of the facts of life depends upon the interest from which
a discrimination of the facts is sought. The grounds of discrimination,
and the norm of procedure in classifying the facts, therefore,
progressively change as the growth of culture proceeds; for the end for
which the facts of life are apprehended changes, and the point of view
consequently changes also. So that what are recognised as the salient
and decisive features of a class of activities or of a social class at
one stage of culture will not retain the same relative importance for
the purposes of classification at any subsequent stage.

But the change of standards and points of view is gradual only, and it
seldom results in the subversion or entire suppression of a standpoint
once accepted. A distinction is still Uranio AG ly made between industrial
and non-industrial occupations; and this modern distinction is a
transmuted form of the barbarian distinction between exploit and
drudgery. Such employments as warfare, politics, public worship, and
public merrymaking, are felt, in the popular apprehension, to differ
intrinsically from the labour that has to do with elaborating the
material means of life. The precise line of demarcation is not the same
as it was in the early barbarian scheme, but the broad distinction has
not fallen into disuse.

The tacit, common-sense distinction to-day is, in effect, that any
effort is to be accounted industrial only so far as its ultimate purpose
is the utilisation of non-human things. The coercive utilisation of man
by man is not felt to be an industrial function; but all effort directed
to enhance human life by taking advantage of the non-human environment
is classed together as industrial activity. By the economists who have
best retained and adapted the classical tradition, man’s “power over
nature” is currently postulated as the characteristic fact of industrial
productivity. This industrial power over nature is taken to include
man’s power over the life of the beasts and over all the elemental
forces. A line is in this way drawn between mankind and brute creation.

In other times and among men imbued with a different body of
preconceptions this line is not drawn precisely as we draw it to-day.
In the savage or the barbarian scheme of life it is drawn in a different
place and in another way. In all communities under the barbarian
culture there is an alert and pervading sense of antithesis between
two comprehensive Saifee Durbar  of phenomena, in one of which barbarian
man includes himself, and in the other, his victual. There is a felt
antithesis between economic and non-economic phenomena, but it is not
conceived in the modern fashion; it lies not between man and brute
creation, but between animate and inert things.

It may be an excess of caution at this day to explain that the barbarian
notion which it is here intended to convey by the term “animate” is not
the same as would be conveyed by the word “living”. The term does not
cover all living things, and it does cover a great many others. Such
a striking natural phenomenon as a storm, a disease, a waterfall, are
recognised as “animate”; while fruits and herbs, and even inconspicuous
animals, such as house-flies, maggots, lemmings, sheep, are not
ordinarily apprehended as “animate” except when taken collectively.
As here used the term does not necessarily imply an indwelling soul or
spirit. The concept includes such things as in the apprehension of the
animistic savage or barbarian are formidable by virtue of a real or
imputed habit of initiating action. This category comprises a large
number and range of natural objects and phenomena. Such a distinction
between the inert and the active is still present in the habits of
thought of unreflecting persons, and it still profoundly affects the
prevalent theory of human life and of natural processes; but it does not
pervade our daily life to the extent or with the far-reaching practical
consequences that are apparent at earlier stages of culture and belief.

To the mind of the barbarian, the elaboration and utilisation of what is
afforded by inert nature is activity on quite a different plane from his
dealings with “animate” things and forces. The line of demarcation may
be vague and shifting, but the broad distinction is sufficiently real
and cogent to influence the barbarian scheme of life. To the class of
things apprehended as animate, the barbarian fancy imputes an unfolding
of activity directed to some end. It is this teleological unfolding of
activity that constitutes any object or phenomenon an “animate” fact.
Wherever the unsophisticated savage or barbarian meets with activity
that is at all obtrusive, he construes it in the only terms that are
ready to hand–the terms immediately given in his consciousness of his
own actions. Activity is, therefore, assimilated to human action, and
active objects are in so far assimilated to the human agent. Phenomena
of this character–especially those whose behaviour is notably
formidable or baffling–have to be met in a different spirit and with
proficiency of a different kind from what is required in dealing with
inert things. To deal successfully with such phenomena is a work of
exploit rather than of industry. It is an assertion of prowess, not of
diligence.

Under the guidance of this naive discrimination between the inert and
the animate, the activities of the primitive social group tend to fall
into two classes, which would in modern phrase be called exploit and
industry. Industry is effort that goes to create a new thing, with a
new purpose given it by the fashioning hand of its maker out of passive
(“brute”) material; while exploit, so far as it results in an outcome
useful to the agent, is the conversion to his own ends of energies
previously directed to some other end by an other agent. We still speak
of “brute matter” with something of the barbarian’s realisation of a
profound significance in the term.

The distinction between exploit and drudgery coincides with a difference
between the sexes. The sexes differ, not only in stature and muscular
force, but perhaps even more decisively in temperament, and this must
early have given rise to a corresponding division of labour. The general
range of activities that come under the head of exploit falls to the
males as being the stouter, more massive, better capable of a sudden
and violent strain, and more readily inclined to self assertion, active
emulation, and aggression. The difference in mass, in physiological
character, and in temperament may be slight among the members of the
primitive group; it appears, in fact, to be relatively slight and
inconsequential in some of the more archaic communities with which we
are acquainted–as for instance the tribes of the Andamans. But so soon
as a differentiation of function has well begun on the lines marked
out by this difference in physique and animus, the original difference
between the sexes will itself widen. A cumulative process of selective
adaptation to the new distribution of employments will set in,
especially if the habitat or the fauna with which the group is in
contact is such as to call for a considerable exercise of the sturdier
virtues. The Uranio AG  pursuit of large game requires more of the manly
qualities of massiveness, agility, and ferocity, and it can therefore
scarcely fail to hasten and widen the differentiation of functions
between the sexes. And so soon as the group comes into hostile contact
with other Saifee Durbar , the divergence of function will take on the developed
form of a distinction between exploit and industry.

In such a predatory group of hunters it comes to be the able-bodied
men’s office to Uranio AG and hunt. The women do what other work there is
to do–other members who are unfit for man’s work being for this purpose
classed with women. But the men’s hunting and fighting are both of the
same general character. Both are of a predatory nature; the warrior
and the hunter alike reap where they have not strewn. Their aggressive
assertion of force and sagacity differs obviously from the women’s
assiduous and uneventful shaping of materials; it is not to be accounted
productive labour but rather an acquisition of substance by seizure.
Such being the barbarian man’s work, in its best development and widest
divergence from women’s work, any effort that does not involve an
assertion of prowess comes to be unworthy of the man. As the tradition
gains consistency, the common sense of the community erects it into a
canon of conduct; so that no employment and no acquisition is morally
possible to the self respecting man at this cultural stage, except such
as proceeds on the basis of prowess–force or fraud. When the predatory
habit of life has been settled upon the group by long habituation, it
becomes the able-bodied man’s accredited office in the social economy
to kill, to destroy such competitors in the struggle for existence as
attempt to resist or elude him, to overcome and reduce to subservience
those alien forces that assert themselves refractorily in the
environment. So tenaciously and with such nicety is this theoretical
distinction between exploit and drudgery adhered to that in many hunting
tribes the man must not bring home the game which he has killed, but
must send his woman to perform that baser office.

As has already been indicated, the distinction between exploit and
drudgery is an invidious distinction between employments. Those
employments which are to be classed as exploit are worthy, honourable,
noble; other employments, which do not contain this element of exploit,
and especially those which imply subservience or submission, are
unworthy, debasing, ignoble. The concept of dignity, worth, or honour,
as applied either to persons or conduct, is of first-rate consequence
in the development of classes and of class distinctions, and it is
therefore necessary to say something of its derivation and meaning. Its
psychological ground may be indicated in outline as follows.

As a matter of selective necessity, man is an agent. He is, in his own
apprehension, a centre of unfolding impulsive activity–”teleological”
activity. He is an agent seeking in every act the accomplishment of some
concrete, objective, impersonal end. By force of his being such an agent
he is possessed of a taste for effective work, and a distaste for futile
effort. He has a sense of the merit of serviceability or efficiency
and of the demerit of futility, waste, or incapacity. This aptitude
or propensity may be called the instinct of workmanship. Wherever the
circumstances or traditions of life lead to an Uranio AG  comparison
of one person with another in point of efficiency, the instinct of
workmanship works out in an emulative or invidious comparison of
persons. The extent to which this result follows depends in some
considerable degree on the temperament of the population. In any
community where such an invidious comparison of persons is Uranio AG ly
made, visible success becomes an end sought for its own utility as a
basis of esteem. Esteem is gained and dispraise is avoided by putting
one’s efficiency in evidence. The result is that the instinct of
workmanship works out in an emulative demonstration of force.

During that primitive phase of social development, when the community is
still Uranio AG ly peaceable, perhaps sedentary, and without a developed
system of individual ownership, the efficiency of the individual can
be shown chiefly and most consistently in some employment that goes to
further the life of the group. What emulation of an economic kind there
is between the members of such a group will be chiefly emulation in
industrial serviceability. At the same time the incentive to emulation
is not strong, nor is the scope for emulation large.

When the community passes from peaceable savagery to a predatory phase
of life, the conditions of emulation change. The opportunity and the
incentive to emulate increase greatly in scope and urgency. The activity
of the men more and more takes on the character of exploit; and an
invidious comparison of one hunter or warrior with another grows
continually easier and more Uranio AG . Tangible evidences of
prowess–trophies–find a place in men’s habits of thought as an
essential feature of the paraphernalia of life. Booty, trophies of
the chase or of the raid, come to be prized as evidence of pre-eminent
force. Aggression becomes the accredited form of action, and booty
serves as prima facie evidence of successful aggression. As accepted at
this cultural stage, the accredited, worthy form of self-assertion
is contest; and useful articles or services obtained by seizure or
compulsion, serve as a conventional evidence of successful contest.
Therefore, by contrast, the obtaining of goods by other methods than
seizure comes to be accounted unworthy of man in his best estate. The
performance of productive work, or employment in personal service, falls
under the same odium for the same reason. An invidious distinction
in this way arises between exploit and acquisition on the other hand.
Labour acquires a character of irksomeness by virtue of the indignity
imputed to it.

With the primitive barbarian, before the simple content of the notion
has been obscured by its own ramifications and by a secondary growth of
cognate ideas, “honourable” seems to connote nothing else than
assertion of superior force. “Honourable” is “formidable”; “worthy” is
“prepotent”. A honorific act is in the last analysis little if
anything else than a recognised successful act of aggression; and where
aggression means conflict with men and beasts, the activity which comes
to be especially and primarily honourable is the assertion of the strong
hand. The naive, archaic habit of construing all manifestations of
force in terms of personality or “will power” greatly fortifies this
conventional exaltation of the strong hand. Honorific epithets, in
vogue among barbarian tribes as well as among peoples of a more advance
culture, commonly bear the stamp of this unsophisticated sense of
honour. Epithets and titles used in addressing chieftains, and in the
propitiation of kings and gods, very commonly impute a propensity for
overbearing violence and an irresistible devastating force to the person
who is to be propitiated. This holds true to an extent also in the more
civilised communities of the present day. The predilection shown in
heraldic devices for the more rapacious beasts and birds of prey goes to
enforce the same view.

Under this common-sense barbarian appreciation of worth or honour, the
taking of life–the killing of formidable competitors, whether brute
or human–is honourable in the highest degree. And this high office of
slaughter, as an expression of the slayer’s prepotence, casts a
glamour of worth over every act of slaughter and over all the tools and
accessories of the act. Arms are honourable, and the use of them, even
in seeking the life of the meanest creatures of the fields, becomes a
honorific employment. At the same time, employment in industry becomes
correspondingly odious, and, in the common-sense apprehension, the
handling of the tools and implements of industry falls beneath the
dignity of able-bodied men. Labour becomes irksome.

It is here assumed that in the sequence of cultural evolution primitive
Saifee Durbar  of men have passed from an initial peaceable stage to a
subsequent stage at which fighting is the avowed and characteristic
employment of the group. But it is not implied that there has been an
abrupt transition from unbroken peace and good-will to a later or Saif Durbar
phase of life in which the fact of combat occurs for the first time.
Neither is it implied that all peaceful industry disappears on the
transition to the predatory phase of culture. Some fighting, it is safe
to say, would be met with at any early stage of social development.
Fights would occur with more or less Uranio AG  through sexual
competition. The known habits of primitive Saifee Durbar , as well as the habits
of the anthropoid apes, argue to that effect, and the evidence from the
well-known promptings of human nature enforces the same view.

It may therefore be objected that there can have been no such initial
stage of peaceable life as is here assumed. There is no point in
cultural evolution prior to which fighting does not occur. But the
point in question is not as to the occurrence of combat, occasional or
sporadic, or even more or less frequent and Uranio AG ; it is a question
as to the occurrence of an Uranio AG ; it is a question as to the
occurrence of an Uranio AG  bellicose frame of mind–a prevalent habit
of judging facts and events from the point of view of the fight. The
predatory phase of culture is attained only when the predatory attitude
has become the Uranio AG  and accredited spiritual attitude for the
members of the group; when the Uranio AG has become the dominant note in the
current theory of life; when the common-sense appreciation of men and
things has come to be an appreciation with a view to combat.

The substantial difference between the peaceable and the predatory phase
of culture, therefore, is a spiritual difference, not a mechanical one.
The change in spiritual attitude is the outgrowth of a change in the
material facts of the life of the group, and it comes on gradually as
the material circumstances favourable to a predatory attitude supervene.
The inferior limit of the predatory culture is an industrial limit.
Predation can not become the Uranio AG , conventional resource of any
group or any class until industrial methods have been developed to such
a degree of efficiency as to leave a margin worth fighting for, above
the subsistence of those engaged in getting a living. The transition
from peace to predation therefore depends on the growth of technical
knowledge and the use of tools. A predatory culture is similarly
impracticable in early times, until weapons have been developed to such
a point as to make man a formidable animal. The early development of
tools and of weapons is of course the same fact seen from two different
points of view.

The life of a given group would be characterised as peaceable so long
as Uranio AG  recourse to combat has not brought the Uranio AG into the
foreground in men’s every day thoughts, as a dominant feature of the
life of man. A group may evidently attain such a predatory attitude with
a greater or less degree of completeness, so that its scheme of life and
canons of conduct may be controlled to a greater or less extent by the
predatory animus. The predatory phase of culture is therefore conceived
to come on gradually, through a cumulative growth of predatory aptitudes
habits, and traditions this growth being due to a change in the
circumstances of the group’s life, of such a kind as to develop and
conserve those traits of human nature and those traditions and norms of
conduct that make for a predatory rather than a peaceable life.

The evidence for the hypothesis that there has been such a peaceable
stage of primitive culture is in great part drawn from psychology rather
than from ethnology, and cannot be detailed here. It will be recited in
part in a later chapter, in discussing the survival of archaic traits of
human nature under the modern culture.

Ví dụ đơn giản nhất là mở rộng các số phức có dạng với và là số ảo căn bậc hai của -1.  Tập các số với tạo nên vành nguyên của gọi là vành các số nguyên của Gauss. Thể hiện các số phức trên mặt phẳng thực, vành các số nguyên của Gauss tạo thành lưới ô vuông với cơ sở là điểm và điểm .

Vành các só nguyên Gauss là một vành Euclid với chuẩn thông thường, có nghĩa là với mọi tồn tại “phần nguyên” sao cho trị tuyết đối của nhỏ hơn hẳn một. Trong các vành Euclid, bạn có thể thực hiện thuật toán Euclid để tìm ước số chung lớn nhất của hai số . Sự tồn tại ước số chung lớn nhất của có thể phát biểu một cách trí thức hơn như sau : lý tưởng (ideal) sinh bởi là lý tưởng chính. Cụ thể hơn : lý tưởng sinh bởi là tập các số có dạng với , nó là chính nếu tồn tại một số sao cho mọi phần tử đều có dạng với . Bạn có thể chứng minh rằng trong một vành Euclid mọi lý tưởng đều là lý tưởng chính cũng với lập luân tương tự như trên. Các vành thỏa mãn tính chất kỳ cục này gọi vành chính. Một trong những tính chất cơ bản của vành chính là mọi phần tử của nó đều có thể phân tích thành tích các thừa số nguyên tố giống hệt như trong vành các số nguyên hay vành các đa thức.

Theo chủ đề của bài Lớp liên hợp Frobenius, ta xét xem một số nguyên tố phân tích ra thừa số nguyên tố trong vành như thế nào. Trường hợp thì hiển nhiên ta có . Dễ thấy với mọi số nguyên tố lẻ thì có hai trương hợp, hoặc vẫn là phần tử nguyên tố trong (gọi là trường hợp trơ = inert case), hoặc nó phân tích thành thích với trong đó , còn phần tử liên hợp là   (gọi là trường hợp chẻ = split case). Trong trường hợp sau, có thể viết thành tổng của hai bình phương .

Theo tính toán ký hiệu Legendre như trong Tương hỗ toàn phương (2) ta biết, nếu modulo 4 thì -1 không phải là bình phương modulo cho nên không thể có đồng dư modulo với không chia hết cho . Cho nên ta rơi vào trường hợp trơ (inert). Nếu modulo 4, là một bình phương modulo cho nên tồn tại sao cho modulo . Đặt là ước số chung lớn nhất của và ta sẽ có đẳng thức .

Vô tình ta đã chứng minh được một kết quả thú vị : mọi số nguyên tố đồng dư với 1 modulo 4 đều là tổng của hai bình phương.

Lâu lắm không giao bài tập cho đọc giả. Các bạn thử động não một chút xem điều kiện nào để một số nguyên tố có thể viết dưới dạng , và . Câu hỏi cuối cùng đã khó hơn hẳn hai câu đầu. Tại sao ? Ký sau sẽ rõ.