151,161 research outputs found
Algebraic Characterization of Uniquely Vertex Colorable Graphs
The study of graph vertex colorability from an algebraic perspective has
introduced novel techniques and algorithms into the field. For instance, it is
known that -colorability of a graph is equivalent to the condition for a certain ideal I_{G,k} \subseteq \k[x_1, ..., x_n]. In this
paper, we extend this result by proving a general decomposition theorem for
. This theorem allows us to give an algebraic characterization of
uniquely -colorable graphs. Our results also give algorithms for testing
unique colorability. As an application, we verify a counterexample to a
conjecture of Xu concerning uniquely 3-colorable graphs without triangles.Comment: 15 pages, 2 figures, print version, to appear J. Comb. Th. Ser.
Structure of sets which are well approximated by zero sets of harmonic polynomials
The zero sets of harmonic polynomials play a crucial role in the study of the
free boundary regularity problem for harmonic measure. In order to understand
the fine structure of these free boundaries a detailed study of the singular
points of these zero sets is required. In this paper we study how "degree
points" sit inside zero sets of harmonic polynomials in of degree
(for all and ) and inside sets that admit
arbitrarily good local approximations by zero sets of harmonic polynomials. We
obtain a general structure theorem for the latter type of sets, including sharp
Hausdorff and Minkowski dimension estimates on the singular set of "degree
points" () without proving uniqueness of blowups or aid of PDE methods
such as monotonicity formulas. In addition, we show that in the presence of a
certain topological separation condition, the sharp dimension estimates improve
and depend on the parity of . An application is given to the two-phase free
boundary regularity problem for harmonic measure below the continuous threshold
introduced by Kenig and Toro.Comment: 40 pages, 2 figures (v2: streamlined several proofs, added statement
of Lojasiewicz inequality for harmonic polynomials [Theorem 3.1]
Limit Theorems for Stochastic Approximations Algorithms With Application to General Urn Models
In the present paper we study the multidimensional stochastic approximation algorithms where the drift function h is a smooth function and where jacobian matrix is diagonalizable over C but assuming that all the eigenvalues of this matrix are in the the region Repzq Ä
0. We give results on the fluctuation of the process around the stable equilibrium point of h. We extend the limit theorem of the one dimensional Robin's Monroe algorithm [MR73]. We give also application of these limit theorem for some class of urn models proving the efficiency of this method
On nonlocal problems for semilinear second order differential inclusions without compactness
Existence of mild solutions for a nonlocal abstract problem driven by a semilinear second order differential inclusion is studied in Banach spaces in the lack of compactness both on the fundamental system generated by the linear part and on the nonlinear multivalued term. The method used for proving our existence theorems is based on the combination of a fixed point theorem and a selection theorem developed by ourselves with an approach that uses De Blasi measure of noncompactness and the weak topology. As application of our existence result we present the study of the controllability of a problem guided by a wave equation
KAJIAN TEOREMA BOLZANO-WEIERSTRASS UNTUK MENGKONSTRUKSI BARISAN YANG KONVERGEN DI R^n DAN APLIKASINYA DALAM PEMBUKTIAN TEOREMA EKSISTENSI MAX-MIN
A sequence is a function from the set of natural numbers to the set of real numbers . In sequences there is the concept of sequence convergence. Testing the convergence of a sequence can be done using the Bolzano-Weierstrass Theorem. This theorem states that every finite sequence has a convergent sequence. The relationship between convergent sequences and finite sequences is also important to study further. Apart from being used to prove the convergence of sequences, the Bolzano-Weirstrass Theorem can also be applied to prove the Max-Min Existence Theorem. This research was conducted to examine the relationship between convergent sequences and finite sequences, the relationship between convergence and continuous functions and the relationship between continuous functions and max-min values ââwith the aim of constructing a convergent sequence in R^n and its application in proving the Max-Min Existence Theorem. This research is a literature study. This research was conducted through a literature review of books and other literature. From the literature review, the materials are then discussed in depth. The results of the literature study show that a convergent sequence is a finite sequence, but a finite sequence is not necessarily convergent. In determining the convergence of a sequence using the Bolzano-Weierstrass Theorem, it is necessary to first show the limitations of the sequence. Furthermore, to prove the Max-Min Existence Theorem it is necessary to require that the sequence is finite and then this theorem can be proven using the Bolzano-Weierstrass Theorem and Apit Theorem.
Keywords: Monotonous Sequence, Finite Sequence, Continuity, Bolzano-Weierstrass Theorem, Max-Min Existence Theorem
Time evolution for the Pauli-Fierz operator (Markov approximation and Rabi cycle)
This article is concerned with a system of particles interacting with the
quantized electromagnetic field (photons) in the non relativistic Quantum
Electrodynamics (QED) framework and governed by the Pauli-Fierz Hamiltonian. We
are interested not only in deriving approximations of several quantities when
the coupling constant is small but also in obtaining different controls of the
error terms. First, we investigate the time dynamics approximation in two
situations, the Markovian (Theorem 1.4 completed by Theorem 1.16) and non
Markovian (Theorem 1.6) cases. These two contexts differ in particular
regarding the approximation leading terms, the error control and the initial
states. Second, we examine two applications. The first application is the study
of marginal transition probabilities related to those analyzed by Bethe and
Salpeter in \cite{B-S}, such as proving the exponential decay in the Markovian
case assuming the Fermi Golden Rule (FGR) hypothesis (Theorem 1.17 or Theorem
1.15) and obtaining a FGR type approximation in the non Markovian case (Theorem
1.5). The second application, in the non Markovian case, includes the
derivation of Rabi cycles from QED (Theorem 1.7). All the results are
established under the following assumptions at some steps of the proofs: an
ultraviolet and an infrared regularization are imposed, the quadratic terms of
the Pauli-Fierz Hamiltonian are dropped, and the dipole approximation is
assumed but only to obtain optimal error controls.Comment: This version improves some results such as Theorem 1.4, which now
includes an estimate without the dipolar approximation, and some proofs are
then reorganize
Central limit theorem for reducible and irreducible open quantum walks
In this work we aim at proving central limit theorems for open quantum walks
on . We study the case when there are various classes of vertices
in the network. Furthermore, we investigate two ways of distributing the vertex
classes in the network. First we assign the classes in a regular pattern.
Secondly, we assign each vertex a random class with a uniform distribution. For
each way of distributing vertex classes, we obtain an appropriate central limit
theorem, illustrated by numerical examples. These theorems may have application
in the study of complex systems in quantum biology and dissipative quantum
computation.Comment: 20 pages, 4 figure
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