Nheyo dzedzidziso dzeboka
Dzidziso yeboka ibazi remasvomhu rinoongorora maumbirwo ealgebraic anozivikanwa semapoka. Mapoka ipfungwa huru mumasvomhu, inoonekwa muzvikamu zvakasiyana-siyana zvakaita sealgebra, geometry, nhamba dzidziso, uye fizikisi. Chinyorwa chino chinangwa ndechekupa pfupiso huru yedzidziso yeboka, tichikurukura tsananguro, mienzaniso, uye mashandisirwo epfungwa yeboka.
Tsanangudzo yeboka
Boka iboka \(G\) rakashongedzerwa nekushanda kwebinary \( \) rinogutsa hunhu huna hunotevera:
1. Kuvhara: Pa \(a, b \mu G\ yega yega), mhedzisiro yekushanda \(ab\) iriwo mu \(G\).
2. Kubatana: Pa \(a, b, c \mu G\ yega yega), inoshanda \((ab) c = a (bc)\).
3. Chinhu Chinosiyanisa Munhu: Pane chinhu \(e \muG\) zvekuti pane chimwe nechimwe \(a \muG\), \(ea = ae = a\) chinoshanda.
4. Chinhu Chinopindirana: Pachinhu chimwe nechimwe \(a \muG\), pane chinhu \(b \muG\) chinoita kuti \(ab = ba = e\), apo \(e\) chiri chinhu chinoratidza hunhu.
Kana seti \(G\) uye mashandiro \( \) zvikateerera zvinhu izvi zvina, ipapo \((G, )\) zvinonzi iboka.
Mienzaniso yeMapoka
Nhamba dzese dzine kuwedzera
Seti yenhamba dzese \(\mathbb{Z}\) ine basa rekuwedzera (\(+\)) inoumba boka.
– Yakavharwa: Kuwedzera nhamba dzese kunoburitsa nhamba yakazara.
– Associative : \((a + b) + c = a + (b + c)\) yezvose \(a, b, c \in \mathbb{Z}\).
– Chinhu cheKuzivikanwa: Chinhu cheKuzivikanwa i0, nekuti \(a + 0 = 0 + a = a\) kune ese \(a \in \mathbb{Z}\).
– Chinhu Chinopindirana: Nhamba yese \(a\) ine chinhu chinopindirana, chinoti \(-a\) nekuti \(a + (-a) = -a + a = 0\).
Nhamba dzese dzeModulo n
Seti \(\mathbb{Z}_n\) ine nhamba \( \{0, 1, …, n-1\} \) ine modulo yekuwedzera \(n\) inoumbawo boka.
– Yakavharwa: Huwandu hwezvinhu \(n\) mu \(\mathbb{Z}_n\) chinhu chiri mu \(\mathbb{Z}_n\).
– Kubatana: Modulo yekuwedzera \(n\) inogutsa hunhu hwekubatana.
- Chinhu cheKuzivikanwa: Chinhu cheKuzivikanwa i0.
– Chinhu Chinopindirana: Pachinhu chimwe nechimwe \(a \mu \mathbb{Z}_n\), chinhu chacho chinopindirana ndi \(na\).
Matrix neMatrix Kuwanza
Seti yemamatrice ese akaenzana \( 2 \times 2 \) asingachinjiki nekushanda kwekuwedzera matrix inoumbawo boka, rinonzi boka guru remutsetse \(GL(2, \mathbb{R})\).
– Yakavharwa: Kuwanda kwemamatrices maviri asingachinjiki kunoita kuti pave nemamatrice asingachinjikiwo.
- Kubatana: Kuwanda kweMatrix kunobatanidza.
– Chinhu cheIdentity: Chinhu cheIdentity ndicho chinoratidza kuti munhu anofanirwa kuva neruzivo, kureva \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.
– Chinhu Chinopinduka: Matrix yega yega isingapinduki ine chinhu chinopinduka, kureva matrix inogutsa \(A \cdot A^{-1} = A^{-1} \cdot A = I\).
Mhando dzeMapoka
Boka reAbelian
Boka reAbelian, kana kuti boka reCommutative, iboka umo mashandiro ebhinari anogutsawo pfuma yecommutative, kureva, \(ab = ba\) ye \(a, b \in G\ yega yega). Mienzaniso yemapoka eAbelian ndeiyi \((\mathbb{Z}, +)\) uye \((\mathbb{R}, +)\).
Boka reKufamba Netsoka
Mapoka eCyclic mapoka anogona kugadzirwa nechinhu chimwe chete. Kureva kuti, pane chinhu \(a \muG\) zvekuti chinhu chega chega chiri mu \(G\) chinogona kunyorwa muchimiro \(a^n\) chenhamba huru \(n\). Muenzaniso weboka reCyclic ndi \((\mathbb{Z}_n, +)\).
Hunhu hweMapoka
Boka diki
Boka diki iboka diki reboka iro ririwo boka rine mashandiro akafanana. Semuenzaniso, seti yenhamba dzakafanana iboka diki reboka renhamba dzese.
Kurongeka kweBoka uye Kurongeka kweElement
Kurongeka kweboka inhamba yezvinhu zviri muboka. Kurongeka kwechinhu \(a \muG\) ndiyo nhamba diki kwazvo \(n\) zvekuti \(a^n = e\).
Mashandisirwo edzidziso yeboka
Dzidziso yeboka ine mashandisirwo akawanda muminda yakasiyana-siyana:
cryptography
Dzidziso yeboka inoshandiswa muma cryptographic algorithms akadai seRSA neDiffie-Hellman zvinoenderana nechimiro cheboka rema modulo numbers.
Dzidziso yeSimmetry
Mufizikisi nekemesitiri, dzidziso yemapoka inoshandiswa kudzidza kuenzana kwemamorekuru nemakristaro. Mapoka ekuenzana anobatsira kuona hunhu hwemuviri nehwemakemikari emamorekuru.
Dzidziso yeGalois
Dzidziso yeboka inoshandiswa mudzidziso yeGalois kudzidza mhinduro dzemaequations epolynomial uye hukama huripo pakati pemidzi yemaequations.
Kugadziriswa kweChiratidzo
Dzidziso yeboka inoshandiswa mukuongorora kweFourier uye kugadzirisa zviratidzo, uko mabasa anoonekwa sezvikamu zvemapoka anoshanda.
Mhedziso
Dzidziso yeboka ibazi guru remasvomhu rine mashandisirwo akapararira munzvimbo dzakasiyana-siyana. Kunzwisisa tsananguro yeboka, mhando dzaro, hunhu hwaro, uye mashandisirwo aro kunopa hwaro hwakasimba hwekutsvaga zvakadzama mumasvomhu nedzimwe sainzi. Nepfungwa dzakadai senhamba dzese uye kuwedzera, matrices uye kuwanda, uye symmetry mumamorekuru, dzidziso yeboka inopa maturusi anodiwa pakugadzirisa matambudziko akasiyana-siyana edzidziso uye anoshanda.
