Searcharxiv⌕ Search

arXiv subjects

Alan Horwitz

Publications and source records attributed to Alan Horwitz.

At least 19 recordsLinked to original sources
arXiv2026

Besant quadrilaterals

We solve the following problem of W.H. Besant using a formula for the coefficients of an ellipse inscribed in a quadrilateral, $Q$: \enquote{If an ellipse be inscribed in a quadrilateral so that one focus is equidistant from the four vertices(call that point $EP$), the other focus must be at the intersection of the diagonals(call that point $IP$).} We also prove somewhat more than just solving Besant's problem itself, though it would be nice to see the details of the geometric approach proposed by Besant. More precisely, we also prove the converse result and additional results when $Q$ is a trapezoid. Finally, we show that such an inscribed ellipse exists if and only if $Q$ is orthodiagonal.

math.HO↗
arXiv2021

A generalization of parallelograms involving inscribed ellipses, conjugate diameters, and tangency chords

A convex quadrilateral, $Q$, is called a midpoint diagonal quadrilateral if the intersection point of the diagonals of $Q$ coincides with the midpoint of at least one of the diagonals of $Q$. A parallelogram, P, is a special case of a midpoint diagonal quadrilateral since the diagonals of P bisect one another. We prove two results about ellipses inscribed in midpoint diagonal quadrilaterals, which generalize properties of ellipses inscribed in parallelograms involving convex quadrilaterals. First, $Q$ is a midpoint diagonal quadrilateral if and only if each ellipse inscribed in $Q$ has tangency chords which are parallel to one of the diagonals of $Q$. Second, $Q$ is a midpoint diagonal quadrilateral if and only if each ellipse inscribed in $Q$ has a pair of conjugate diameters parallel to the diagonals of $Q$. Finally, we show that there is a unique ellipse, $E_I$, of minimal eccentricity incribed in a midpoint diagonal quadrilateral, $Q$, and we show that the equal conjugate diameters of $E_I$ are parallel to the diagonals of $Q$.

math.MG↗
arXiv2021

Midpoint Diagonal Quadrilaterals

A convex quadrilateral, $Q$, is called a midpoint diagonal quadrilateral if the intersection point of the diagonals of $Q$ coincides with the midpoint of at least one of the diagonals of $Q$. A parallelogram, P, is a special case of a midpoint diagonal quadrilateral since the diagonals of P bisect one another. We prove two results about ellipses inscribed in midpoint diagonal quadrilaterals, which generalize properties of ellipses inscribed in parallelograms involving convex quadrilaterals. First, $Q$ is a midpoint diagonal quadrilateral if and only if each ellipse inscribed in $Q$ has tangency chords which are parallel to one of the diagonals of $Q$. Second, $Q$ is a midpoint diagonal quadrilateral if and only if each ellipse inscribed in $Q$ has a unique pair of conjugate diameters parallel to the diagonals of $Q$. Finally, we show that there is a unique ellipse, $E_I$, of minimal eccentricity inscribed in a midpoint diagonal quadrilateral, $Q$, and also that the unique pair of conjugate diameters parallel to the diagonals of $Q$ are the equal conjugate diameters of $E_I$.

math.MG↗
arXiv2017

When is an ellipse inscribed in a quadrilateral tangent at the midpoint of two or more sides ?

In "Quartic Coincidences and the Singular Value Decomposition" by Clifford and Lachance, Mathematics Magazine, December, 2013, it was shown that if there is a midpoint ellipse(an ellipse inscribed in a quadrilateral, $Q$, which is tangent at the midpoints of all four sides of $Q$), then $Q$ must be a parallelogram. We strengthen this result by showing that if $Q$ is not a parallelogram, then there is no ellipse inscribed in $Q$ which is tangent at the midpoint of three sides of $Q$. Second, the only quadrilaterals which have inscribed ellipses tangent at the midpoint of even two sides of $Q$ are trapezoids or what we call a midpoint diagonal quadrilateral(the intersection point of the diagonals of $Q$ coincides with the midpoint of at least one of the diagonals of $Q$).

math.HO↗
arXiv2016

Ellipses of minimal eccentricity inscribed in midpoint diagonal quadrilaterals

In an earlier paper of the author, we showed that there is a unique ellipse of minimal eccentricity, $E_I$, inscribed in any convex quadrilateral, $Q$. Using a different approach in this paper, we prove that there is a unique ellipse of minimal eccentricity, $E_I$, inscribed in a midpoint diagonal quadrilateral, $Q$, which is a quadrilateral with the property that the intersection point of the diagonals of $Q$ coincides with the midpoint of at least one of the diagonals of $Q$. Our main result is that if $Q$ is a midpoint diagonal quadrilateral, then the smallest non-negative angle between equal conjugate diameters of $E_I$ equals the smallest non-negative angle between the diagonals of $Q$. This was proven in another earlier paper of the author for parallelograms.

math.CA↗
arXiv2015

A ladder ellipse problem

We consider a problem similar to the well-known ladder box problem, but where the box is replaced by an ellipse. A ladder of a given length, $s$, with ends on the positive x and y axes, is known to touch an ellipse that lies in the first quadrant and is tangent to the positive x and y axes. We then want to find the height of the top of the ladder above the floor. We show that there is a value, $s = s_0$, such that there is only one possible position of the ladder, while if $s > s_0$, then there are two different possible positions of the ladder. Our solution involves solving an equation which is equivalent to solving a 4th degree polynomial equation.

math.CA↗
arXiv2015

Dynamics of ellipses inscribed in triangles

Suppose that we are given two distinct points, $P_1$ and $P_2$, in the interior of a triangle, $T$. Is there always an ellipse inscribed in $T$ which also passes through $P_1$ and $P_2$ ? If yes, how many such ellipses ? We answer those questions in this paper. It turns out that, except for $P_1$ and $P_2$ on a union of three line segments, there are four such ellipses which pass through $P_1$ and $P_2$. We also answer a similar question if instead $P_1$ and $P_2$ lie on the boundary of $T$. Finally, an interesting related question, is the following: Given a point, $P$, in the interior of a triangle, $T$, and a real number, $r$, is there always an ellipse inscribed in $T$ which passes through $P$ and has slope $r$ at $P$ ? Again, if yes, how many such ellipses ? The answer is somewhat different than for the two point case without specifying a slope. There are cases where no such ellipse exists.

math.CA↗
arXiv2015

Dynamics of ellipses inscribed in quadrilaterals

Let Q be a convex quadrilateral in the xy plane and let int(Q) denote the interior of Q. Let D_1 and D_2 denote the diagonals of Q and let P denote their point of intersection. For (i)-(iii), let P_0 = (x_0,y_0) be a point in the interior of Q. We prove the following: (i) If P_0 does not lie on D_1 or on D_2, then there are exactly two ellipses inscribed in Q which pass through P_0. (ii) If P_0 does lie on D_1 or on D_2, but does not equal P, then there is exactly one ellipse inscribed in Q which passes through P_0. (iii) There is no ellipse inscribed in Q which passes through P. (iv) If P_0 lies on the boundary of Q, but P_0 is not one of the vertices of Q, then there is exactly one ellipse inscribed in Q which passes through P_0(and is thus tangent to Q at one of its sides).

math.CA↗
arXiv2015

Means and non-real Intersection Points of Taylor Polynomials

Suppose that f has continuous derivatives thru order r+1 for x>0, and let P_{c} denote the Taylor polynomial to f of order r at x=c,c>0. In a previous paper of the author, it was shown that if r is an odd whole number and the (r+1)st derivative of f is nonzero on [a,b], then there is a unique x_{0},a<x_{0}<b, such that P_{a}(x_{0})=P_{b}(x_{0}). This defines a mean, depending on f and r, given by m(a,b)=x_{0}. In this paper we discuss the real parts of the pairs of complex conjugate non-real roots of P_{b}-P_{a}. We prove some results for r in general, but our most significant results are for the case r=3. We prove in that case that if f(z)=z^{p}, where p is an integer, p not equal to 0,1,2, or 3, then P_{b}-P_{a} has non-real roots with real part strictly between a and b for any 0<a<b. This defines a countable family of means. We construct a cubic polynomial, g, whose real root gives the real part of the pair of complex conjugate non-real roots of P_{b}-P_{a}. Instead of working directly with a formula for the roots of a cubic, we use the Intermediate Value Theorem to show that g has a root in (a,b).

math.CA↗
arXiv2014

A logarithmic mean and intersections of osculating hyperplanes

We discuss a special case of a family of means defined using intersections of osculating hyperplanes to curves in R^n. Let C be the curve in R^n with vector equation x_{k}(t)=t(ln t)^{k-1},k=1,...,n. Let O_{k} be the osculating hyperplane to C at a_{k},k=1,...,n. Then we show that O_{1},...,O_{n} have a unique point of intersection, P=(i_{1},...,i_{n}), and in particular, i_{1} equals the logarithmic mean in n variables of Neuman.

math.CA↗
arXiv2013

Weighted Quasi-Arithmetic Means and Invariance of Types 1, 2, and 3

Let m_{n} and m_{n-1} be an n mean and an n-1 mean, respectively, n\geq3. If x=(x_{1},...,x_{n}), let π_{\neqj}x=(x_{1},...,x_{j-1},x_{j+1},...,x_{n}). m_{n-1} and m_{n} are said to form a type 1 invariant pair if m_{n}(m_{n-1}(π_{\neq1}x),m_{n-1}(π_{\neq2}x),...,m_{n-1}(π_{\neqn}x))=m_{n}(x) for all x\inR^{n}. m_{n-1} and m_{n} are said to form a type 2 invariant pair if m_{n}(x,m_{n-1}(x))=m_{n-1}(x) for all x\inR_{+}^{n-1}. If x=(x_{1},...,x_{n-1}), let π_{=j}x=(x_{1},...,x_{j-1},x_{j},x_{j},x_{j+1},...,x_{n-1})\inR_{+}^{n}. m_{n-1} and m_{n} are said to form a type 3 invariant pair if m_{n-1}(m_{n}(π=_{1}x),...,m_{n}(π_{=n-1}x))=m_{n-1}(x) for all x\inR_{+}^{n-1}. Let m_{h,w,n}(a_{1},...,a_{n})=h^{-1}(((\sum_{k=1}^{n}w(a_{k})h(a_{k}))/(\sum_{k=1}^{n}w(a_{k})))), where h(x) is continuous and monotone, and w(x) is continuous and positive, on (0,\infty) denote the family of weighted quasi--arithmetic means in n variables. We prove that if m_{h,w,n} and m_{h,w,n-1} form a type 1 or type 3 invariant pair, then m_{h,w,n} and m_{h,w,n-1} are quasi--arithmetic means. The method of proof involves deriving equations for certain partial derivatives of order 3 of m_{h,w,n} on the diagonal of R_{+}^{n}. The proof also requires an equation relating certain partial derivatives of order 3 for type 1 or type 3 invariant pairs of means. We also show that any pair of weighted quasi--arithmetic means m_{h,w,n} and m_{h,w,n-1} form a type 2 invariant pair.

math.CA↗
arXiv2011

Illumination by Tangent Lines

Let f be a differentiable function on the real line, and let P\inG_{f}^{C}= all points not on the graph of f. We say that the illumination index of P, denoted by I_{f}(P), is k if there are k distinct tangents to the graph of f which pass through P. In section 2 we prove results about the illumination index of f with f" (x)\geq 0 on \Re. In particular, suppose that y=L_1(x) and y=L_2(x) are distinct oblique asymptotes of f and let P=(s,t)\in G_{f}^{C}. If max(L_1(s),L_2(s))<t<f(s), then I_{f}(P)=2. If L_1(s)\not= L_2(s) and min(L_1(s),L_1(s))<t\leqmax(L_1(s),L_2(s)), then I_{f}(P)=1. Finally, if t_\leqmin(L_1(s),L_2(s)), then I_{f}(P)=0. We also show that any point below the graph of a convex rational function or exponential polynomial must have illumination index equal to 2. In section 3 we also prove results about the illumination index of polynomials.

math.CA↗
arXiv2009

An Area Inequality for Ellipses Inscribed in Quadrilaterals

If E is any ellipse inscribed in a convex quadrilateral, D, then we prove that Area(E)/Area(D) is less than or equal to pi/4, and equality holds if and only if D is a parallelogram and E is tangent to the sides of D at the midpoints. This extends well known results for ellipses inscribed in triangles. We also prove that the foci of the unique ellipse of maximal area inscribed in a parallelogram, D, lie on the orthogonal least squares line for the vertices of D. This does not hold in general for convex quadrilaterals.

math.CA↗
arXiv2008

Ellipses Inscribed in Parallelograms

We prove that there exists a unique ellipse of minimal eccentricity, E_{I}, inscribed in a parallelogram, D. We also prove that the smallest nonnegative angle between equal conjugate diameters of E_{I} equals the smallest nonnegative angle between the diagonals of D. We also prove that if E_{M} is the unique ellipse inscribed in a rectangle, R, which is tangent at the midpoints of the sides of R, then E_{M} is the unique ellipse of minimal eccentricity, maximal area, and maximal arc length inscribed in R. Let D be any convex quadrilateral. In previous papers, the author proved that there is a unique ellipse of minimal eccentricity, E_{I}, inscribed in D, and a unique ellipse, E_{O}, of minimal eccentricity circumscribed about D. We defined D to be bielliptic if E_{I} and E_{O} have the same eccentricity. In this paper we show that a parallelogram, D, is bielliptic if and only if the square of the length of one of the diagonals of D equals twice the square of the length of one of the sides of D.

math.MG↗
arXiv2008

Means and Hermite Interpolation

Let $m_{2}<m_{1}$ be two given nonnegative integers with $n=m_{1}+m_{2}+1$. For suitably differentiable $f$, we let $P,Q\in π_{n}$ be the Hermite polynomial interpolants to $f$ which satisfy $P^{(j)}(a)=f^{(j)}(a),j=0,1,...,m_{1}$ and $P^{(j)}(b)=f^{(j)}(b),j=0,1,...,m_{2},$ $Q^{(j)}(a)=f^{(j)}(a),j=0,1,...,m_{2}$ and $Q^{(j)}(b)=f^{(j)}(b),j=0,1,...,m_{1}$. Suppose that $f\in C^{n+2}(I)$ with $f^{(n+1)}(x)\neq 0$ for $x\in (a,b)$. If $m_{1}-m_{2}$ is even, then there is a unique $x_{0},a<x_{0}<b,$ such that $P(x_{0})=Q(x_{0})$. If $m_{1}-m_{2}$ is odd, then there is a unique $x_{0},a<x_{0}<b,$ such that $f(x_{0})=\tfrac{1}{2}(P(x_{0})+Q(x_{0})) $. $x_{0}$ defines a strict, symmetric mean, which we denote by $M_{f,m_{1},m_{2}}(a,b)$. We prove various properties of these means. In particular, we show that $f(x)=x^{m_{1}+m_{2}+2}$ yields the arithmetic mean, $f(x)=x^{-1}$ yields the harmonic mean, and $f(x)=x^{(m_{1}+m_{2}+1)/2}$ yields the geometric mean.

math.CA↗
arXiv2007

Ellipses of minimal area and of minimal eccentricity circumscribed about a convex quadrilateral

First, we fill in key gaps in Steiner's nice characterization of the most nearly circular ellipse which passes through the vertices of a convex quadrilateral, D. Steiner proved that there is only one pair of conjugate directions, M1 and M2, that belong to all ellipses of circumscription. Then he proves that if there is an ellipse, E, whose equal conjugate diameters possess the directional constants M1 and M2, then E must be an ellipse of circumscription which has minimal eccentricity. However, Steiner does not show the existence or uniqueness of such an ellipse. We prove that there is a unique ellipse of minimal eccentricity which passes through the vertices of D. We also show that there exists an ellipse which passes through the vertices of D and whose equal conjugate diameters possess the directional constants M1 and M2. We also show that there exists a unique ellipse of minimal area which passes through the vertices of D. Finally, we call a convex quadrilateral, D, bielliptic if the unique inscribed and circumscribed ellipses of minimal eccentricity have the same eccentricity. This generalizes the notion of bicentric quadrilaterals. In particular we show the existence of a bielliptic convex quadrilateral which is not bicentric.

math.CA↗
arXiv2007

Complex Ratios of Cubic Polynomials

Let $p(w)=(w-w_{1})(w-w_{2})(w-w_{3}),$with $\func{Re}w_{1}<\func{Re}w_{2}<\func{Re}w_{3}$. Assume that if the critical points of $p$ are not identical, then they cannot have equal real parts. Define the ratios $σ_{1}=\dfrac{z_{1}-w_{1}}{w_{2}-w_{1}}$ and $σ_{2}=\dfrac{z_{2}-w_{2}}{w_{3}-w_{2}}$. $(σ_{1},σ_{2})$ is called the \QTR{it}{ratio vector} of $p$. This extends the definition of ratio vectors given in earlier papers for polynomials of degree $n$ with all real roots. We then derive bounds on the real part, imaginary part, and modulus of the ratios and also some relations between the ratios. In particular, we prove that $\func{Re}σ_{1}\leq \func{Re}σ_{2}$. We also show that the ratios are real if and only if the roots of $p$ are collinear.

math.CV↗