Mostrando recursos 1 - 20 de 694

  1. Proof of the planar double bubble conjecture using metacalibration methods

    Dorff, Rebecca; Lawlor, Gary; Sampson, Donald; Wilson, Brandon
    We prove the double bubble conjecture in [math] : that the standard double bubble in [math] is boundary length-minimizing among all figures that separately enclose the same areas. Our independent proof is given using the new method of metacalibration, a generalization of traditional calibration methods useful in minimization problems with fixed volume constraints.

  2. Some results on the size of sum and product sets of finite sets of real numbers

    Hart, Derrick; Niziolek, Alexander
    Let [math] and [math] be finite subsets of positive real numbers. Solymosi gave the sum-product estimate [math] , where [math] is the ceiling function. We use a variant of his argument to give the bound ¶ max ( | A + B | , | A B | ) ( 4 log | A | log | B | ) 1 3 | A | 2 3 | B | 2 3 . ¶ (This isn’t quite a generalization since the logarithmic losses are worse here than in Solymosi’s bound.) ¶ Suppose that [math] is a finite subset of real numbers. We show...

  3. Frame theory for binary vector spaces

    Bodmann, Bernhard; Le, My; Reza, Letty; Tobin, Matthew; Tomforde, Mark
    We develop the theory of frames and Parseval frames for finite-dimensional vector spaces over the binary numbers. This includes characterizations which are similar to frames and Parseval frames for real or complex Hilbert spaces, and the discussion of conceptual differences caused by the lack of a proper inner product on binary vector spaces. We also define switching equivalence for binary frames, and list all equivalence classes of binary Parseval frames in lowest dimensions, excluding cases of trivial redundancy.

  4. A tiling approach to Fibonacci product identities

    Artz, Jacob; Rowell, Michael
    In 1998 Filipponi and Hart introduced a number of Fibonacci product identities. This paper provides a combinatorial proof for such identities via tilings. The methods used in the proof are then further used to produce some new Zeckendorf representations and a known Fibonacci identity.

  5. Ineffective perturbations in a planar elastica

    Peterson, Kaitlyn; Manning, Robert
    An elastica is a bendable one-dimensional continuum, or idealized elastic rod. If such a rod is subjected to compression while its ends are constrained to remain tangent to a single straight line, buckling can occur: the elastic material gives way at a certain point, snapping to a lower-energy configuration. ¶ The bifurcation diagram for the buckling of a planar elastica under a load [math] is made up of a trivial branch of unbuckled configurations for all [math] and a sequence of branches of buckled configurations that are connected to the trivial branch at pitchfork bifurcation points. We use several perturbation expansions...

  6. Markov partitions for hyperbolic sets

    Fisher, Todd; Rathnakumara, Himal
    We show that if [math] is a diffeomorphism of a manifold to itself, [math] is a mixing (or transitive) hyperbolic set, and [math] is a neighborhood of [math] , then there exists a mixing (or transitive) hyperbolic set [math] with a Markov partition such that [math] . We also show that in the topologically mixing case the set [math] will have a unique measure of maximal entropy.

  7. Isometric composition operators acting on the Chebyshev space

    Goebeler, Thomas; Potter, Ashley
    Norms of certain composition operators are given in terms of their symbols in some finite-dimensional setting. Then a family of isometric composition operators acting on certain vector spaces is identified.

  8. Symbolic computation of degree-three covariants for a binary form

    Hagedorn, Thomas; Wilson, Glen
    We use elementary linear algebra to explicitly calculate a basis for, and the dimension of, the space of degree-three covariants for a binary form of arbitrary degree. We also give an explicit basis for the subspace of covariants complementary to the space of degree-three reducible covariants.

  9. On the orbits of an orthogonal group action

    Czarnecki, Kyle; Howe, R. Michael; McTavish, Aaron
    Let [math] be the Lie group [math] and let [math] be the vector space of [math] real matrices. An action of [math] on [math] is given by ¶ ( g , h ) . v : = g 1 v h , ( g , h ) G , v V . ¶ We consider the orbits of this group action and demonstrate a cross-section to the orbits. We then determine the stabilizer for a typical element in this cross-section and completely describe the fundamental group of an orbit of maximal dimension.

  10. On the consistency of finite difference approximations of the Black–Scholes equation on nonuniform grids

    Baker, Myles; Sheng, Daniel
    The Black–Scholes equation has been used for modeling option pricing extensively. When the volatility of financial markets creates irregularities, the model equation is difficult to solve numerically; for this reason nonuniform grids are often used for greater accuracy. This paper studies the numerical consistency of popular explicit, implicit and leapfrog finite difference schemes for solving the Black–Scholes equation when nonuniform meshes are utilized. Mathematical tools including Taylor expansions are used throughout our analysis. The consistency ensures the basic reliability of the finite difference schemes based on choices of temporal and variable spatial derivative approximations. Truncation error terms are derived and...

  11. Some numerical radius inequalities for Hilbert space operators

    Omidvar, Mohsen Erfanian; Sal Moslehian, Mohammad; Niknam, Asdolah
    We present several numerical radius inequalities for Hilbert space operators. More precisely, we prove that if [math] and [math] then [math] and [math] . We also show that if [math] is positive, then ¶ w ( A X X A ) 1 2 A ( X + X 2 1 2 ) .

  12. Geometric properties of Shapiro–Rudin polynomials

    Benedetto, John; Sugar Moore, Jesse
    The Shapiro–Rudin polynomials are well traveled, and their relation to Golay complementary pairs is well known. Because of the importance of Golay pairs in recent applications, we spell out, in some detail, properties of Shapiro–Rudin polynomials and Golay complementary pairs. However, the theme of this paper is an apparently new elementary geometric observation concerning cusp-like behavior of certain Shapiro–Rudin polynomials.

  13. Minimum spanning trees

    Jayawant, Pallavi; Glavin, Kerry
    The minimum spanning tree problem originated in the 1920s when O. Borůvka identified and solved the problem during the electrification of Moravia. This graph theory problem and its numerous applications have inspired many others to look for alternate ways of finding a spanning tree of minimum weight in a weighted, connected graph since Borůvka’s time. This note presents a variant of Borůvka’s algorithm that developed during the graph theory course work of undergraduate students. We discuss the proof of the algorithm, compare it to existing algorithms, and present an implementation of the procedure in Maple.

  14. Newton's law of heating and the heat equation

    Gockenbach, Mark; Schmidtke, Kristin
    Newton’s law of heating models the average temperature in an object by a simple ordinary differential equation, while the heat equation is a partial differential equation that models the temperature as a function of both space and time. A series solution of the heat equation, in the case of a spherical body, shows that Newton’s law gives an accurate approximation to the average temperature if the body is not too large and it conducts heat much faster than it gains heat from the surroundings. Finite element simulation confirms and extends the analysis.

  15. A complete classification of $\mathbb{Z}_{p}$-sequences corresponding to a polynomial

    Huang, Leonard
    Let [math] be a prime number and set [math] . A [math] -sequence is a function [math] . Let [math] be the set [math] . We prove that the set of sequences of the form [math] , where [math] , is precisely the set of periodic [math] -sequences with period equal to a [math] -power. Given a [math] -sequence, we will also determine all [math] that correspond to the sequence according to the manner above.

  16. Yet another generalization of frames and Riesz bases

    Joveini, Reza; Amini, Massoud
    A frame is a sequence of vectors in a Hilbert space satisfying certain inequalities that make it valuable for signal processing and other purposes. There is a formula giving the reconstruction of a signal (a vector in the space) from its sequence of inner products (the Fourier coefficients) with the elements of the frame sequence. A [math] -frame, or operator-valued frame, is a sequence of operators defined on a countable ordered index set that has properties analogous to those of a frame sequence. ¶ We present a new approach to the matter of defining a Hilbert space frame, indexed by an...

  17. Contributions to Seymour's second neighborhood conjecture

    Brantner, James; Brockman, Greg; Kay, Bill; Snively, Emma
    Let [math] be a simple digraph without loops or digons. For any [math] let [math] be the set of all nodes at out-distance 1 from [math] and let [math] be the set of all nodes at out-distance 2. We show that if the underlying graph is triangle-free, there must exist some [math] such that [math] . We provide several properties a “minimal” graph which does not contain such a node must have. Moreover, we show that if one such graph exists, then there exist infinitely many.

  18. Automatic growth series for right-angled Coxeter groups

    Glover, Rebecca; Scott, Richard
    Right-angled Coxeter groups have a natural automatic structure induced by their action on a CAT( [math] ) cube complex. The normal form for this structure is defined with respect to the generating set consisting of all cliques in the defining graph for the group. In this paper we study the growth series for right-angled Coxeter groups with respect to this automatic generating set. In particular, we show that there exist nonisomorphic Coxeter groups with the same automatic growth series, and give a comparison with the usual growth series defined with respect to the standard generating set.

  19. Construction and enumeration of Franklin circles

    Garcia, Rebecca; Meyer, Stefanie; Sanders, Stacey; Seitz, Amanda
    Around 1752, Benjamin Franklin constructed a variant on the popular magic squares and what we call a magic [math] -circle. We provide a definition for magic [math] -circles, magic [math] -circles and, more specifically, Franklin magic [math] -circles. In this paper, we use techniques in computational algebraic combinatorics and enumerative geometry to construct and to count Franklin magic 8-circles. We also provide a description of its minimal Hilbert basis and determine the symmetry operations on Franklin magic 8-circles.

  20. Bifurcus semigroups and rings

    Adams, Donald; Ardila, Rene; Hannasch, David; Kosh, Audra; McCarthy, Hanah; Ponomarenko, Vadim; Rosenbaum, Ryan
    A bifurcus semigroup or ring is defined as possessing the strong property that every nonzero nonunit nonatom may be factored into two atoms. We develop basic properties of such objects as well as their relationships to well-known semigroups and rings.

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