Uniqueness of Steiner minimal trees on boundaries in general position

Abstract

In order to examine the effect of pressure on the anharmonicity of Au, extended x-ray absorption fine-structure spectra near the Au L 3 edge were measured in the temperature range from 300 to 1100 K under pressures up to 14 GPa using large-volume high-pressure devices and synchrotron radiation. The anharmonic effective pair potentials of Au, V (u) = au 2 /2! + bu 3 /3!, at 0.1 MPa, 6 and 14 GPa have been calculated. The pressure dependence of the thermal expansion coefficients has also been evaluated. The reliability of the anharmonic correction proposed on the basis of the Anderson scale has been discussed.

On the configuration space of Steiner minimal trees

“…This paper continues the study of the decomposition of the configuration space of n points in R d as defined by the combinatorial types of the Steiner minimal trees, which was initiated in [11]. We consider the ordered setting and our main result is a proof that the cells consisting of configurations with unambiguous Steiner minimal trees are path-connected.…”

“…For example, it is not known that the space of ambiguous Steiner minimal trees has measure zero. The only partial result is in d = 2 dimensions, where the unambiguous Steiner minimal trees contain an everywhere dense open subset of the configuration space, but the proof in [11] is long and complicated. The approach in [14] leads to a shorter proof.…”

Current Open Problems in Discrete and Computational Geometry

“…The global structure is not well understood. Generally, a boundary set permits several shortest networks -for example, the vertices of the square in the plane -but as it is proved in [36], a finite subset of the plane "in general position" (i.e. for an open everywhere dense set of n-element subsets of the plane) permits unique shortest tree.…”

“…Notice that the proof in [36] is based on the local structure of the shortest tree and on the geometry of the plane. Another proof suggested in [37] is more topological, deals with a wider class of locally minimal networks [32], but it cannot be extended to the general case.…”

Το πρόβλημα Fermat-Torricelli και ένα αντίστροφο πρόβλημα στο Κ-επίπεδο και σε κλειστά πολύεδρα του R^3

“…και a 30 ′ = a 3 (βλέπε σχήμα 6.1). Αναφέρουμε ότι η μοναδικότητα της λύσης του πλήρους δέντρου Steiner για κυρτά τετράπλευρα σε ένα Κεπίπεδο έχει μελετηθεί (για κυρτό σύνορο σε μία πολλαπλότητα Riemann) στις εργασίες [41], [42] και [45]. Θα επικεντρωθούμε στην επίλυση του γενικευμένου προβλήματος Gauss στο Κ-επίπεδο, όπου θα δείξουμε ότι η δυναμική πλαστικότητα των κυρτών τετραπλεύρων αίρεται, αν θεωρήσουμε ένα πλήρες δέντρο Steiner που αποτελείται από δύο γενικευμένα σημεία Fermat-Torricelli (b.FT) στο εσωτερικό του κυρτού τετραπλεύ- έτσι ώστε:…”