| 1. |
- Alexandersson, Per, 1987-, et al.
(author)
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Discriminants, Symmetrized Graph monomials and Sums of Squares
- 2012
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In: Experimental Mathematics. - : Informa UK Limited. - 1058-6458 .- 1944-950X. ; 21:4, s. 353-361
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Journal article (peer-reviewed)abstract
- In 1878, motivated by the requirements of the invariant the-ory of binary forms, J. J. Sylvester constructed, for every graphwith possible multiple edges but without loops, its symmetrizedgraph monomial, which is a polynomial in the vertex labels ofthe original graph. We pose the question for which graphs thispolynomial is nonnegative or a sum of squares. This problem ismotivated by a recent conjecture of F. Sottile and E. Mukhin onthe discriminant of the derivative of a univariate polynomial andby an interesting example of P. and A. Lax of a graph with fouredges whose symmetrized graph monomial is nonnegative butnot a sum of squares. We present detailed information about sym-metrized graph monomials for graphs with four and six edges,obtained by computer calculations.
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| 2. |
- Alexandersson, Per
(author)
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Polytopes and Large Counterexamples
- 2019
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In: Experimental Mathematics. - : TAYLOR & FRANCIS INC. - 1058-6458 .- 1944-950X. ; 28:1, s. 115-120
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Journal article (peer-reviewed)abstract
- In this short note, we give large counterexamples to natural questions about certain order polytopes, in particular, Gelfand–Tsetlin polytopes. Several of the counterexamples are too large to be discovered via a brute-force computer search. We also show that the multiset of hooks in a Young diagram is not enough information to determine the Ehrhart polynomial for an associated order polytope. This is somewhat counter-intuitive to the fact that the multiset of hooks always determine the leading coefficient of the Ehrhart polynomial.
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| 3. |
- Amerik, Ekaterina, et al.
(author)
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Evidence for the Dynamical Brauer-Manin Criterion
- 2016
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In: Experimental Mathematics. - : Taylor & Francis. - 1058-6458 .- 1944-950X. ; 25:1, s. 54-65
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Journal article (peer-reviewed)abstract
- Let phi: X -> X be a morphism of a variety over a number field K. We consider local conditions and a "Brauer-Manin" condition, defined by Hsia and Silverman, for the orbit of a point P is an element of X(K) to be disjoint from a subvariety V subset of X, i.e., for V boolean AND O-phi (P) = empty set. We provide evidence that the dynamical Brauer-Manin condition is sufficient to explain the lack of points in the intersection V boolean AND O-phi (P); this evidence stems from a probabilistic argument as well as unconditional results in the case of etale maps.
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| 4. |
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| 5. |
- Avelin, Helen
(author)
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Numerical Computations of Green's Function and Its Fourier Coefficients on PSL(2, Z)
- 2010
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In: Experimental Mathematics. - 1058-6458 .- 1944-950X. ; 19:3, s. 335-343
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Journal article (peer-reviewed)abstract
- We present some examples of numerical investigations of the value distribution of Green's function and of its Fourier coefficients on the modular group PSL(2,Z) Our results indicate that both Green's function G(s)(z, w) and its Fourier coefficients F-n(z,s) have a Gaussian value distribution in the semiclassical limit when Re s = 1/2.
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| 6. |
- Belova, Anna, et al.
(author)
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Critical points of the multiplier map for the quadratic family
- 2022
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In: Experimental Mathematics. - : Informa UK Limited. - 1058-6458 .- 1944-950X. ; 31:1, s. 337-345
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Journal article (peer-reviewed)abstract
- The multiplier λn of a periodic orbit of period n can be viewed as a (multiple-valued) algebraic function on the space of all complex quadratic polynomials pc(z)=z2+c. We provide a numerical algorithm for computing critical points of this function (i.e., points where the derivative of the multiplier with respect to the complex parameter c vanishes). We use this algorithm to compute critical points of λn up to period n = 10.
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| 7. |
- Bergström, Jonas, et al.
(author)
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Eisenstein Congruences for SO(4, 3), SO(4, 4), Spinor, and Triple Product L-values
- 2018
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In: Experimental Mathematics. - : Informa UK Limited. - 1058-6458 .- 1944-950X. ; 27:2, s. 230-250
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Journal article (peer-reviewed)abstract
- We work out instances of a general conjecture on congruences between Hecke eigenvalues of induced and cuspidal automorphic representations of a reductive group, modulo divisors of certain critical L-values, in the case that the group is a split orthogonal group. We provide some numerical evidence in the case that the group is SO(4, 3) and the L-function is the spinor L-function of a genus 2, vector-valued, Siegel cusp form. We also consider the case that the group is SO(4, 4) and the L-function is a triple product L-function.
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| 8. |
- Bernhardsson, Bo, et al.
(author)
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Singular Values of Trilinear Forms
- 2001
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In: Experimental Mathematics. - : Informa UK Limited. - 1944-950X .- 1058-6458. ; 10:4, s. 509-518
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Journal article (peer-reviewed)abstract
- Let T : H 1 × H 2 × H 3 → C be a trilinear form, where H 1, H 2, H 3 are separable Hilbert spaces. In the hypothesis that at least two of the three spaces are finite dimensional we show that the norm square λ = ∥T∥2 is a root of a certain algebraic equation, usually of very high degree, which we baptize the millennia] equation, because it is an analogue of the secular equation in the bilinear case. More generally, as indicated in the title, we can consider singular values of a trilinear form and their squares too satisfy the same equation. We work out the binary case (all three spacesare two dimensional). Even in this case the situation is complex, so, in the absence of any genuine results, we content ourselves with advancing a number of conjectures suggested by computer experiments. Finally, we connect the singular values of a trilinear form with the critical values of an associated family of a one parameter family of bilinear forms. Also here we have to offer mainly only experimental evidence.
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| 9. |
- Björner, Anders., et al.
(author)
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Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincare homology 3-sphere
- 2000
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In: Experimental Mathematics. - 1058-6458 .- 1944-950X. ; 9:2, s. 275-289
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Journal article (peer-reviewed)abstract
- We present a computer program based on bistellar operations that provides a useful too I for the construction of simplicial manifolds with few vertices. As an example, we obtain a 16-vertex triangulation of the Poincare homology 3-sphere; we construct an infinite series of non-FL d-dimensional spheres with d + 13 vertices for d greater than or equal to 5; and we show that ifa d-manifold, with d greater than or equal to 5, admits any triangulation on n vertices, it admits a noncombinatorial triangulation on n + 12 vertices.
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| 10. |
- Boege, Tobias
(author)
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No Eleventh Conditional Ingleton Inequality
- 2023
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In: Experimental Mathematics. - : Informa UK Limited. - 1058-6458 .- 1944-950X. ; , s. 1-9
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Journal article (peer-reviewed)abstract
- A rational probability distribution on four binary random variables (Formula presented.) is constructed which satisfies the conditional independence relations (Formula presented.) but whose entropy vector violates the Ingleton inequality. This settles a recent question of Studený (IEEE Trans. Inf. Theory vol. 67, no. 11) and shows that there are, up to symmetry, precisely ten inclusion-minimal sets of conditional independence assumptions on four discrete random variables which make the Ingleton inequality hold. The last case in the classification of which of these inequalities are essentially conditional is also settled.
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