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• 11:14, 24 January 2025 Hbghlyj talk contribs created page Talk:Irrational Number Space is Complete Metric Space (Alternative proof is to show that the irrationals are homeomorphic to $ω^ω$ via continued fractions: new section)
• 10:49, 28 October 2024 Hbghlyj talk contribs uploaded File:1.svg (<code>\begin{tikzpicture} \fill[red!50](.8,0)--plot[domain=.8:1/1.7](\x,1/\x)-|(0,0); \draw(0,0)rectangle(.8,1.7); \node at(.8,0)[below]{$a$}; \node at(0,1.7)[left]{$b$}; \draw[->] (-0.2,0)--(2.2,0) node[right] {$x$}; \draw[->] (0,-0.2)--(0,2.7) node[above] {$y$}; \draw[color=blue,domain=.4:2.5] plot (\x,1/\x); \node at(1,1.2)[right]{$y=x^{p-1}$}; \end{tikzpicture}</code>)
• 10:49, 28 October 2024 Hbghlyj talk contribs created page File:1.svg (<code>\begin{tikzpicture} \fill[red!50](.8,0)--plot[domain=.8:1/1.7](\x,1/\x)-|(0,0); \draw(0,0)rectangle(.8,1.7); \node at(.8,0)[below]{$a$}; \node at(0,1.7)[left]{$b$}; \draw[->] (-0.2,0)--(2.2,0) node[right] {$x$}; \draw[->] (0,-0.2)--(0,2.7) node[above] {$y$}; \draw[color=blue,domain=.4:2.5] plot (\x,1/\x); \node at(1,1.2)[right]{$y=x^{p-1}$}; \end{tikzpicture}</code>)
• 10:45, 28 October 2024 Hbghlyj talk contribs uploaded File:Reverse Holder's Ineq Proof Integral 2.svg (<code>\begin{tikzpicture} \fill[red!50](1/1.7,1.7)plot[domain=1/1.7:.4](\x,1/\x)-|(0,1.7); \draw(0,0)rectangle(.8,1.7); \node at(.8,0)[below]{$a$}; \node at(0,1.7)[left]{$b$}; \draw[->] (-0.2,0)--(2.2,0) node[right] {$x$}; \draw[->] (0,-0.2)--(0,2.7) node[above] {$y$}; \draw[color=blue,domain=.4:2.5] plot (\x,1/\x); \node at(1,1.2)[right]{$x=y^{-q-1}$}; \end{tikzpicture}</code>)
• 10:45, 28 October 2024 Hbghlyj talk contribs created page File:Reverse Holder's Ineq Proof Integral 2.svg (<code>\begin{tikzpicture} \fill[red!50](1/1.7,1.7)plot[domain=1/1.7:.4](\x,1/\x)-|(0,1.7); \draw(0,0)rectangle(.8,1.7); \node at(.8,0)[below]{$a$}; \node at(0,1.7)[left]{$b$}; \draw[->] (-0.2,0)--(2.2,0) node[right] {$x$}; \draw[->] (0,-0.2)--(0,2.7) node[above] {$y$}; \draw[color=blue,domain=.4:2.5] plot (\x,1/\x); \node at(1,1.2)[right]{$x=y^{-q-1}$}; \end{tikzpicture}</code>)
• 10:40, 28 October 2024 Hbghlyj talk contribs uploaded File:Reverse Holder's Ineq Proof Integral 1.svg (<code>\begin{tikzpicture} \fill[red!50](.8,0)--(.8,1.7)plot[domain=.8:.4](\x,1/\x)--(0,1/.4)--(0,0)--(.8,0); \draw(0,0)rectangle(.8,1.7); \node at(.8,0)[below]{$a$}; \node at(0,1.7)[left]{$b$}; \draw[->] (-0.2,0)--(2.2,0) node[right] {$x$}; \draw[->] (0,-0.2)--(0,2.7) node[above] {$y$}; \draw[color=blue,domain=.4:2.5] plot (\x,1/\x); \node at(1,1.2)[right]{$y=x^{p-1}$}; \end{tikzpicture}</code>)
• 10:40, 28 October 2024 Hbghlyj talk contribs created page File:Reverse Holder's Ineq Proof Integral 1.svg (<code>\begin{tikzpicture} \fill[red!50](.8,0)--(.8,1.7)plot[domain=.8:.4](\x,1/\x)--(0,1/.4)--(0,0)--(.8,0); \draw(0,0)rectangle(.8,1.7); \node at(.8,0)[below]{$a$}; \node at(0,1.7)[left]{$b$}; \draw[->] (-0.2,0)--(2.2,0) node[right] {$x$}; \draw[->] (0,-0.2)--(0,2.7) node[above] {$y$}; \draw[color=blue,domain=.4:2.5] plot (\x,1/\x); \node at(1,1.2)[right]{$y=x^{p-1}$}; \end{tikzpicture}</code>)
• 10:30, 28 October 2024 Hbghlyj talk contribs uploaded File:Reverse Holder's Ineq.svg (<code>\begin{tikzpicture} \draw(0,0)rectangle(.8,1.7); \node at(.8,0)[below]{$a$}; \node at(0,1.7)[left]{$b$}; \draw[->] (-0.2,0)--(2.2,0) node[right] {$x$}; \draw[->] (0,-0.2)--(0,2.2) node[above] {$y$}; \draw[color=blue,domain=.4:2.5] plot (\x,1/\x); \node at(1,1.2)[right]{$y=x^{p-1}$}; \end{tikzpicture}</code>)
• 10:30, 28 October 2024 Hbghlyj talk contribs created page File:Reverse Holder's Ineq.svg (<code>\begin{tikzpicture} \draw(0,0)rectangle(.8,1.7); \node at(.8,0)[below]{$a$}; \node at(0,1.7)[left]{$b$}; \draw[->] (-0.2,0)--(2.2,0) node[right] {$x$}; \draw[->] (0,-0.2)--(0,2.2) node[above] {$y$}; \draw[color=blue,domain=.4:2.5] plot (\x,1/\x); \node at(1,1.2)[right]{$y=x^{p-1}$}; \end{tikzpicture}</code>)
• 19:14, 9 May 2024 Hbghlyj talk contribs uploaded File:LorDeltaSequence2.png
• 19:14, 9 May 2024 Hbghlyj talk contribs created page File:LorDeltaSequence2.png
• 19:04, 9 May 2024 Hbghlyj talk contribs created page N over 2 times Reciprocal of 1 Plus n Squared x Squared to the Power of 3/2 Delta Sequence (Created page with "== Theorem == <onlyinclude> Let $\sequence {\map {\delta_n} x}$ be a sequence such that: :$\ds \map {\delta_n} x := \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} }$ Then $\sequence {\map {\delta_n} x}_{n \mathop \in {\N_{>0} } }$ is a delta sequence. That is, in the distributional sense it holds that: :$\ds \lim_{n \mathop \to \infty} \map {\delta_n} x = \map \delta x$ or :$\ds \lim_{n \m...")
• 00:20, 4 May 2024 Hbghlyj talk contribs created page Vector Space over an Infinite Field is not equal to the Union of Proper Subspaces (add a theorem needed in the proof of primitive element theorem)
• 23:25, 3 May 2024 Hbghlyj talk contribs created page Talk:Separable Degree of Field Extensions is Multiplicative (Created page with "== About different definitions == I read from [https://math.stackexchange.com/questions/591077/multiplicative-nature-of-the-separability-degree stackexchange]: If one starts with separability degree definition 2, as done in the book by Lang, the result follows without much difficulty. But it is intractable if one starts from definition 1 --~~~~")
• 22:56, 3 May 2024 Hbghlyj talk contribs created page Equivalence of Definitions of Separable Degree (Add proof)
• 20:26, 3 May 2024 Hbghlyj talk contribs created page Steinitz's Theorem (A theorem needed in the proof of Primitive Element Theorem. name based on Wikipedia page https://en.wikipedia.org/wiki/Steinitz%27s_theorem_(field_theory))
• 19:53, 3 May 2024 Hbghlyj talk contribs created page Separable Degree of Field Extensions is Multiplicative (Creating the page based on a theorem in Lang's book, which in needed in Transitivity of Separable Field Extensions)
• 19:51, 3 May 2024 Hbghlyj talk contribs created page Separable Degree is At Most Equal To Degree (Creating the page based on a theorem in Lang's book, which in needed in Transitivity of Separable Field Extensions)
• 19:37, 3 May 2024 Hbghlyj talk contribs created page Definition:Separable Degree/Definition 3 (Created page with "== Definition == <onlyinclude> Let $K$ be a normal extension of $F$ that contains $E$. The '''separable degree''' $\index E F_{\operatorname {sep} }$ of $E / F$ is the number of embeddings of $E$ into $K$ that fix $F$. </onlyinclude> Category:Separable Field Extensions")
• 19:35, 3 May 2024 Hbghlyj talk contribs created page Definition:Separable Degree/Definition 2 (Created page with "== Definition == <onlyinclude> Let $\bar F$ be the algebraic closure of $F$. The '''separable degree''' $\index E F_{\operatorname {sep} }$ of $E / F$ is the number of embeddings of $E$ into $\bar F$ that fix $F$. </onlyinclude> Category:Separable Field Extensions")
• 19:32, 3 May 2024 Hbghlyj talk contribs created page Definition:Separable Degree/Definition 1 (Created page with "== Definition == <onlyinclude> Let $S \subseteq E$ be the separable closure of $F$ in $E$. The '''separable degree''' $\index E F_{\operatorname {sep} }$ of $E / F$ is the degree $\index S F$. </onlyinclude> Category:Separable Field Extensions")
• 19:01, 3 May 2024 Hbghlyj talk contribs created page Transitivity of Separable Field Extensions (Create the page which is a non-existent link in Subextensions of Separable Field Extension are Separable)
• 16:14, 3 May 2024 Hbghlyj talk contribs created page Equivalence of Definitions of Purely Inseparable Extension (Created page with "== Theorem == Let $E/F$ be an algebraic field extension. {{TFAE|def = Purely Inseparable Field Extension}} === Definition 1 === {{:Definition:Purely Inseparable Field Extension/Definition 1}} === Definition 2 === {{:Definition:Purely Inseparable Field Extension/Definition 2}} === Definition:Purely In...")
• 16:13, 3 May 2024 Hbghlyj talk contribs created page Definition:Purely Inseparable Field Extension/Definition 3 (Created page with "== Definition == <onlyinclude> Let $F$ have positive characteristic $p$. The extension $E/F$ is '''purely inseparable''' {{iff}} each element of $E$ has a minimal polynomial of the form $X^{p^n} - a$. </onlyinclude> == Also see == * Equivalence of Definitions of Purely Inseparable Extension Category:Definitions/Field Extensions")
• 16:12, 3 May 2024 Hbghlyj talk contribs created page Definition:Purely Inseparable Field Extension/Definition 2 (Created page with "== Definition == <onlyinclude> Let $F$ have positive characteristic $p$. The extension $E/F$ is '''purely inseparable''' {{iff}} for each $\alpha \in E$ there exists $n \in \N$ such that $\alpha^{p^n} \in F$. </onlyinclude> == Also see == * Equivalence of Definitions of Purely Inseparable Extension Category:Definitions/Field Extensions")
• 16:11, 3 May 2024 Hbghlyj talk contribs created page Definition:Purely Inseparable Field Extension/Definition 1 (Created page with "== Definition == <onlyinclude> The extension $E/F$ is '''purely inseparable''' {{iff}} every element $\alpha \in E \setminus F$ is inseparable. </onlyinclude> == Also see == * Equivalence of Definitions of Purely Inseparable Extension Category:Definitions/Field Extensions")
• 12:03, 1 May 2024 Hbghlyj talk contribs created page Group of Units Ring of Integers Modulo p^2 is Cyclic (Created page with "== Theorem == Let $p$ be a prime. Let $\struct {\Z / p^2 \Z, +, \times}$ be the ring of integers modulo $p^2$. Let $U = \struct {\paren {\Z / p^2 \Z}^\times, \times}$ denote the group of units of $\struct {\Z / p^2 \Z, +, \times}$. Then $U$ is cyclic. == Proof == The case $p = 2$ follows from Isomorp...")
• 11:00, 1 May 2024 Hbghlyj talk contribs created page Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n/Lemma (Created page with "=== Lemma === There are only $5$ numbers $n$ with the property that $\map \phi n = 8$, and they are $15$, $16$, $20$, $24$ and $30$. === Proof of lemma === Let $p$ be a prime factor of $n$. By Euler Phi Function is Multiplicative: :$p - 1 = \map \phi p \divides \map \phi n = 8$ so $p \in \set{2, 3, 5}$. Let $i,j,k \in \Z^{\ge 0}$ such that $n = 2^i 3^j 5^k$. By Euler Phi Function is Multiplicative: :$8 = \map \phi n = \map \phi {...")
• 16:03, 21 April 2024 Hbghlyj talk contribs created page Talk:Ideal Contained in Finite Union of Prime Ideals (Other name?: new section)
• 12:24, 12 April 2024 Hbghlyj talk contribs created page Template talk:Journalref (in example?: new section)
• 18:04, 25 March 2024 Hbghlyj talk contribs created page User talk:Sandbox/Du Bois-Reymond Constants/Example/First (Proving the inequality (1): new section)
• 13:05, 24 March 2024 Hbghlyj talk contribs created page User talk:Hbghlyj/Sandbox/James's Theorem (Add a redirect?: new section)
• 12:50, 24 March 2024 Hbghlyj talk contribs created page James's theorem (Created page with "== Theorem == $X$ is a Banach space and its dual space is $X^*$. $X$ is reflexive {{iff}} for all $f \in X^*$, there exists an element $a \in X$ such that :$\norm a \leq 1$ and $\map f a = \norm f$ == Proof == === Sufficient Condition === From a corollary of Hahn-Banach Theorem: :For any nonzero element $z \in V$, : :there exists a Definition:Bounded Linear Functional...")
• 12:13, 24 March 2024 Hbghlyj talk contribs created page Reflexive Riesz Lemma (Created page with "== Theorem == <onlyinclude> Let $X$ be a reflexive normed vector space. Let $Y$ be a proper closed linear subspace of $X$. Then there exists $x_\alpha \in X$ such that: :$\norm {x_\alpha} = 1$ with: :$\inf_{y \in M} \norm {\bar{x}-y} = 1$ for all $y \in Y$. </onlyinclude> == Proof == By Hahn-Banach Theorem, there exists $f \in...")
• 21:12, 23 March 2024 Hbghlyj talk contribs created page File talk:Fontene2.gif (thumbnail not animated: new section)
• 20:57, 23 March 2024 Hbghlyj talk contribs created page Talk:Fontené Theorems/Second (Figure updated: new section)
• 20:51, 23 March 2024 Hbghlyj talk contribs uploaded a new version of File:Fontene2.gif (high resolution)
• 20:34, 23 March 2024 Hbghlyj talk contribs created page Talk:Second Fontené Theorem (Duplicate page?: new section)
• 18:40, 23 March 2024 Hbghlyj talk contribs created page Definition:Isogonal Conjugate (Created page with "== Definition == $X$ is a point in the plane of the triangle $ABC$. Then the reflection of the lines $AX$, $BX$, and $CX$ about the angle bisectors at $A$, $B$, and $C$ concur at the '''isogonal conjugate''' of $X$. == Sources == * {{MathWorld|Isogonal Conjugate|IsogonalConjugate}} Category:Triangles")
• 18:33, 23 March 2024 Hbghlyj talk contribs created page Fontené Theorems/Third (Created page with "== Theorem == <onlyinclude> Denote the isogonal conjugate of $P$ with respect to triangle $A B C$ as $P'$. $O$ is the circumcenter. Then the pedal circle of $P$ is tangent to the nine point circle of triangle $A B C$ {{iff}} $O, P, P'$ are collinear. </onlyinclude> == Proof == By Fontené Theorems/Second we can prove that the second intersection $Q'$ of the Defin...")
• 18:28, 23 March 2024 Hbghlyj talk contribs created page Fontené Theorems/Second (Created page with "== Theorem == <onlyinclude> thumb|center|second Fontené theorem If a point $P$ moves on a fixed line $d$ through the circumcenter $O$, then its pedal circle passes through a fixed point $F$ on the nine point circle as illustrated above. </onlyinclude> == Proof == By Fontené Theorems/First, the point of contact $Q$ of the circle $E$ and the ci...")
• 18:24, 23 March 2024 Hbghlyj talk contribs created page Fontené Theorems/First (Created page with "== Theorem == <onlyinclude> Given triangle $ABC$. Let $P$ be an arbitrary point in the plane. $A_1, B_1, C_1$ are the midpoints of $B C, C A, A B$ $A_2 B_2 C_2$ is the pedal triangle of $P$ with respect to triangle $A B C$. Let $X, Y, Z$ be the intersections of $B_1 C_1$ and $B_2 C_2, A_1 C_1$ and $A_2 C_2, A_1 B_1$ and $A_2 B_2$. Then $A_2 X, B_2 Y, C_2 Z$ concur...")
• 17:49, 23 March 2024 Hbghlyj talk contribs created page Definition talk:Mean Curvature (Different concepts?: new section)
• 17:23, 23 March 2024 Hbghlyj talk contribs created page C^k Function Space is Banach Space (Created page with "== Theorem == Let $I = \closedint a b$ be a closed real interval. Let $\struct {\map {C^k} I, +, \, \cdot \,}_\R$ be the vector space of real-valued functions, k-times differentiable on $I$. Let $x \in \map {C^k} I$ be a real-valued function of differentiability class $k$. Let $\norm {...")
• 16:42, 23 March 2024 Hbghlyj talk contribs created page Talk:If Definite Integral of a(x)h(x) vanishes for any C^0 h(x) then C^0 a(x) vanishes (Other names: new section)
• 15:33, 23 March 2024 Hbghlyj talk contribs created page Talk:Fontené Theorems (Other name of the second Fontené theorem?: new section)
• 15:25, 23 March 2024 Hbghlyj talk contribs created page Definition:Pedal Circle (Created page with "== Definition == <onlyinclude> Let $\triangle ABC$ be a triangle. Let $P$ be a point in the plane of $\triangle ABC$. Let $PD$, $PE$ and $PF$ be perpendiculars dropped from $P$ to $BC$, $AC$ and $AB$ respectively. Let $\triangle DEF$ be the pedal triangle of $P$ with respect to $\triangle ABC$. The Definition:Circumcircle...")
• 15:10, 23 March 2024 Hbghlyj talk contribs created page Fontené Theorems (Created page with "== Theorem == thumb|center|second Fontené theorem The second Fontené theorem states that if a point moves on a fixed line through the circumcenter, then its pedal circle passes through a fixed point on the nine point circle as illustrated above. == Sources == * {{MathWorld|Fontené Theorems|FonteneTheorems}} Category:Triangles Category:Ci...")
• 15:09, 23 March 2024 Hbghlyj talk contribs uploaded File:Fontene2.gif
• 15:09, 23 March 2024 Hbghlyj talk contribs created page File:Fontene2.gif