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1.
Among the discrete evolution equations describing a quantum system ℋ
S
undergoing repeated quantum interactions with a chain of exterior systems, we study and characterize those which are directed
by classical random variables in ℝ
N
. The characterization we obtain is entirely algebraical in terms of the unitary operator driving the elementary interaction.
We show that the solutions of these equations are then random walks on the group U(ℋ0) of unitary operators on ℋ0. 相似文献
2.
Torbj?rn Tambour 《Arkiv f?r Matematik》1991,29(1):127-182
In this thesis, we consider some aspects ofnoncommutative classical invariant theory, i.e., noncommutative invariants ofthe classical group SL(2, k). We develop asymbolic method for invariants and covariants, and we use the method to compute some invariant algebras. The subspaceĨ
d
m
of the noncommutative invariant algebraĨ
d
consisting of homogeneous elements of degreem has the structure of a module over thesymmetric group S
m
. We find the explicit decomposition into irreducible modules. As a consequence, we obtain theHilbert series of the commutative classical invariant algebras. TheCayley—Sylvester theorem and theHermite reciprocity law are studied in some detail. We consider a new power series H(Ĩ
d,t) whose coefficients are the number of irreducibleS
m
-modules in the decomposition ofĨ
d
m
, and show that it is rational. Finally, we develop some analogues of all this for covariants. 相似文献
3.
The classification of extended affine Lie algebras of type A_1 depends on the Tits-Kantor- Koecher (TKK) algebras constructed from semilattices of Euclidean spaces.One can define a unitary Jordan algebra J(S) from a semilattice S of R~v (v≥1),and then construct an extended affine Lie algebra of type A_1 from the TKK algebra T(J(S)) which is obtained from the Jordan algebra J(S) by the so-called Tits-Kantor-Koecher construction.In R~2 there are only two non-similar semilattices S and S′,where S is a lattice and S′is a non-lattice semilattice.In this paper we study the Z~2-graded automorphisms of the TKK algebra T(J(S)). 相似文献
4.
A. V. Komlov 《Theoretical and Mathematical Physics》2007,153(3):1643-1651
We consider a Grassmannian version of the noncommutative U(1) sigma model specified by the energy functional E(P) = ‖[a, P]‖
HS
2
, where P is an orthogonal projection operator in a Hilbert space H and a: H → H is the standard annihilation operator. With
H realized as a Bargmann-Fock space, we describe all solutions with a one-dimensional range and prove that the operator [a,
P] is densely defined in H for a certain class of projection operators P with infinite-dimensional ranges and kernels.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 347–357, December, 2007. 相似文献
5.
V. K. Oikonomou 《Theoretical and Mathematical Physics》2011,166(3):337-355
We calculate the scalar Casimir energy and Casimir force for an ℝ3 × N Kaluza-Klein piston setup in which the extra-dimensional space N contains a noncommutative two-dimensional sphere S
fz. We study the cases with T
d
×S
fz and S
fz as the extra-dimensional spaces, where T
d is the d-dimensional commutative torus, and examine the validity of the results and the regularization obtained in the piston setup
in each case. We examine the Casimir energy with one-loop corrections for one piston chamber due to the self-interacting scalar
field in the noncommutative geometry. We compute with some approximations. We compare the obtained results with the results
of analogous computations for the Minkowski space-time M
D
. In conclusion, we discuss the stabilization of the extra-dimensional space in the piston setup. 相似文献
6.
We consider the moment space Mn\mathcal{M}_{n} corresponding to p×p real or complex matrix measures defined on the interval [0,1]. The asymptotic properties of the first k components of a uniformly distributed vector (S1,n, ... , Sn,n)* ~ U (Mn)(S_{1,n}, \dots , S_{n,n})^{*} \sim\mathcal{U} (\mathcal{M}_{n}) are studied as n→∞. In particular, it is shown that an appropriately centered and standardized version of the vector (S
1,n
,…,S
k,n
)∗ converges weakly to a vector of k independent p×p Gaussian ensembles. For the proof of our results, we use some new relations between ordinary moments and canonical moments
of matrix measures which are of their own interest. In particular, it is shown that the first k canonical moments corresponding to the uniform distribution on the real or complex moment space Mn\mathcal{M}_{n} are independent multivariate Beta-distributed random variables and that each of these random variables converges in distribution
(as the parameters converge to infinity) to the Gaussian orthogonal ensemble or to the Gaussian unitary ensemble, respectively. 相似文献
7.
We show that the gauge invariance of the operator ∫ dx tr(A
μ
2
−2/(gξ)x
υθμυ
A
μ) in a noncommutative gauge theory does not lead to the gauge independence of its vacuum condensate. We obtain the generalized
Ward identities for Green’s functions containing the operator limΩ→∞(1/Ω)∫Ω
dx tr (A
μ
2
) in commutative and noncommutative gauge theories.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 3, pp. 350–356, September, 2006. 相似文献
8.
Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aw of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr(Aw)=1/d.tr(Aρ).It is proved that the set of all C^*-algebras of sections of locally trivial C^*-algebra bundles over S^2 with fibres Aω has a group sturcture,denoted by π1^s(Aut(Aω)),which is isomorphic to Zif Ed>1 and {0} if d>1.Let Bcd be a cd-homogeneous C^*-algebra over S^2×T^2 of which no non-trivial matrix algebra can be factored out.The spherical noncommutative torus Sρ^cd is defined by twisting C^*(T2×Z^m-2) in Bcd ×C^*(Z^m-3) by a totally skew multiplier ρ on T^2×Z^m-2。It is shown that Sρ^cd×Mρ∞ is isomorphic to C(S^2)×C^*(T^2×Z^m-2,ρ)× Mcd(C)×Mρ∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p. 相似文献
9.
Palle E. T. Jorgensen 《Israel Journal of Mathematics》1986,56(2):129-142
LetH be an infinite-dimensional separable Hilbert space, and letS=(S
ij)∈teB(H)⊗M
2 be a unitary 2 × 2 matrix with operator entries. We study theC*-algebra generated by the operatorsS
ij, and show that the study of unitary dilations of isometriesT inH reduces to the special case whereS
11 =T, andS
21 = 0. We useC*-algebraic techniques to obtain detailed results about the set of all unitary dilations ofT.
Work supported in part by NSF. 相似文献
10.
Let ℝℝ denote the set of real valued functions defined on the real line. A map D: ℝℝ → ℝℝ is said to be a difference operator if there are real numbers a
i, b
i (i = 1, …, n) such that (Dƒ)(x) = ∑
i=1
n
a
i
ƒ(x + b
i) for every ƒ ∈ ℝℝand x ∈ ℝ. By a system of difference equations we mean a set of equations S = {D
i
ƒ = g
i: i ∈ I}, where I is an arbitrary set of indices, D
i is a difference operator and g
i is a given function for every i ∈ I, and ƒ is the unknown function. One can prove that a system S is solvable if and only if every finite subsystem of S is solvable. However, if we look for solutions belonging to a given class of functions then the analogous statement is no
longer true. For example, there exists a system S such that every finite subsystem of S has a solution which is a trigonometric polynomial, but S has no such solution; moreover, S has no measurable solutions.
This phenomenon motivates the following definition. Let
be a class of functions. The solvability cardinal sc(
) of
is the smallest cardinal number κ such that whenever S is a system of difference equations and each subsystem of S of cardinality less than κ has a solution in
, then S itself has a solution in
. In this paper we determine the solvability cardinals of most function classes that occur in analysis. As it turns out, the
behaviour of sc(
) is rather erratic. For example, sc(polynomials) = 3 but sc(trigonometric polynomials) = ω
1, sc({ƒ: ƒ is continuous}) = ω
1 but sc({f : f is Darboux}) = (2
ω
)+, and sc(ℝℝ) = ω. We consistently determine the solvability cardinals of the classes of Borel, Lebesgue and Baire measurable functions, and
give some partial answers for the Baire class 1 and Baire class α functions.
Partially supported by Hungarian Scientific Foundation grants no. 49786,37758,F 43620 and 61600.
Partially supported by Hungarian Scientific Foundation grant no. 49786. 相似文献
11.
M. F. Gamal’ 《Journal of Mathematical Sciences》2010,165(4):435-448
In 2005, the following question was posed by Duggal, Djordjević, and Kubrusly: Assume that T is a contraction of the class C
10 such that I − T
*
T is compact and the spectrum of T is the unit disk. Can the isometric asymptote of T be a reductive unitary operator? In this paper, we give a positive answer to this question. We construct two kinds of examples.
One of them are the operators of multiplication by independent variable in the closure of analytic polynomials in L
2(ν),where ν is an appropriate positive finite Borel measure on the closed unit disk. The second kind of examples is based on a theorem
by Chevreau, Exner, and Pearcy. We obtain a contraction T satisfying all the needed conditions and such that I − T
*
T belongs to the Schatten–von Neumann classes
\mathfrakSp {\mathfrak{S}_p} for all p > 1. We give an example of a contraction T such that I − T
*
T belongs to
\mathfrakSp {\mathfrak{S}_p} for all p > 1, T is quasisimilar to a unitary operator and has “more” invariant subspaces than this unitary operator. Also, following Bercovici
and Kérchy, we show that if a subset of the unit circle is the spectrum of a contraction quasisimilar to a given absolutely
continuous unitary operator, then this contraction T can be chosen so that I − T*T is compact. Bibliography: 29 titles. 相似文献
12.
Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<(
n
n+x
). Here we prove that the order x Veronese embedding ofP
n
is not weakly (k−1)-defective, i.e. for a general S⊃P
n
such that #(S) = k+1 the projective space | I
2S
(x)| of all degree t hypersurfaces ofP
n
singular at each point of S has dimension (
n
/n+x
)−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I
2S
(x)| has an ordinary double point at each P∈ S and Sing (F)=S.
The author was partially supported by MIUR and GNSAGA of INdAM (Italy). 相似文献
13.
For a domainU on a certaink-dimensional minimal submanifold ofS
n orH
n, we introduce a “modified volume”M(U) ofU and obtain an optimal isoperimetric inequality forU k
k
ω
k
M (D)
k-1
≤Vol(∂D)
k
, where ω
k
is the volume of the unit ball ofR
k
. Also, we prove that ifD is any domain on a minimal surface inS
+
n
(orH
n, respectively), thenD satisfies an isoperimetric inequality2π A≤L
2+A2 (2π A≤L2−A2 respectively). Moreover, we show that ifU is ak-dimensional minimal submanifold ofH
n, then(k−1) Vol(U)≤Vol(∂U).
Supported in part by KME and GARC 相似文献
14.
15.
Klaus Schiefermayr 《Constructive Approximation》2011,33(3):425-432
Let S be a compact infinite set in the complex plane with 0∉S, and let R
n
be the minimal residual polynomial on S, i.e., the minimal polynomial of degree at most n on S with respect to the supremum norm provided that R
n
(0)=1. For the norm L
n
(S) of the minimal residual polynomial, the limit k(S):=limn?¥n?{Ln(S)}\kappa(S):=\lim_{n\to\infty}\sqrt[n]{L_{n}(S)} exists. In addition to the well-known and widely referenced inequality L
n
(S)≥κ(S)
n
, we derive the sharper inequality L
n
(S)≥2κ(S)
n
/(1+κ(S)2n
) in the case that S is the union of a finite number of real intervals. As a consequence, we obtain a slight refinement of the Bernstein–Walsh
lemma. 相似文献
16.
LetW be an algebraically closed filed of characteristic zero, letK be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value, and letA(K) (resp. ℳ(K)) be the set of entire (resp. meromorphic) functions inK. For everyn≥7, we show that the setS
n(b) of zeros of the polynomialx
n−b (b≠0) is such that, iff, g ∈W[x] or iff, g ∈A(K), satisfyf
−1(S
n(b))=g
−1(S
n(b)), thenf
n=g
n. For everyn≥14, we show thatS
n(b) is such that iff, g ∈W({tx}) or iff, g ∈ ℳ(K) satisfyf
−1(S
n(b))=g
−1(S
n(b)), then eitherf
n=g
n, orfg is a constant. Analogous properties are true for complex entire and meromorphic functions withn≥8 andn≥15, respectively.
For everyn≥9, we show that the setY
n(c) of zeros of the polynomial
, (withc≠0 and 1) is an ursim ofn points forW[x], and forA(K). For everyn≥16, we show thatY
n(c) is an ursim ofn points forW(x), and for ℳ(K). We follow a method based on thep-adic Nevanlinna Theory and use certain improvement of a lemma obtained by Frank and Reinders. 相似文献
17.
Boris Rubin 《Journal d'Analyse Mathématique》1999,77(1):105-128
Explicit inversion formulas are obtained for the hemispherical transform(FΜ)(x) = Μ{y ∃S
n :x. y ≥ 0},x ∃S
n, whereS
n is thendimensional unit sphere in ℝn+1,n ≥ 2, and Μ is a finite Borel measure onS
n. If Μ is absolutely continuous with respect to Lebesgue measuredy onS
n, i.e.,dΜ(y) =f(y)dy, we write(F f)(x) = ∫
x.y> 0
f(y)dy and consider the following cases: (a)f ∃C
∞(Sn); (b)f ∃ Lp(S
n), 1 ≤ p < ∞; and (c)f ∃C(Sn). In the case (a), our inversion formulas involve a certain polynomial of the Laplace-Beltrami operator. In the remaining
cases, the relevant wavelet transforms are employed. The range ofF is characterized and the action in the scale of Sobolev spacesL
p
γ
(Sn) is studied. For zonalf ∃ L1(S
2), the hemispherical transformF f was inverted explicitly by P. Funk (1916); we reproduce his argument in higher dimensions.
Partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation
(Germany). 相似文献
18.
Andrei Heilper 《Israel Journal of Mathematics》1979,34(1-2):1-11
Let {Zn=1{(
n
∓
) bea sequence of points in the unit open disk, and letNϕ(U) denote the class of functionsf analytic in the unit disk U such that |f|∈L (
ϕ
1
)(U). For ϕ ≡ 1, the necessary and sufficient conditions for the existence off εN(U) and vanishing atz
n is Σ(
n=1
∓
) (1–|Zn|)2 ∞. Also we estimate a large family of canonical products. These results are extended to ϕ(z)=(1-|z|)ϕ.
This represents a part of a Ph.D. thesis conducted at the Technion — Israel Institute of Technology, Department of Mathematics,
by Dr. C. A. Horowitz. His help during the preparation of this paper is gratefully acknowledged. 相似文献
19.
Vladimir A. Kozlov 《Arkiv f?r Matematik》1999,37(2):305-322
The equationx
(n)(t)=(−1)
n
│x(t)│
k
withk>1 is considered. In the casen≦4 it is proved that solutions defined in a neighbourhood of infinity coincide withC(t−t0)−n/(k−1), whereC is a constant depending only onn andk. In the general case such solutions are Kneser solutions and can be estimated from above and below by a constant times (t−t
0)−n/(k−1). It is shown that they do not necessarily coincide withC(t−t0)−n/(k−1). This gives a negative answer to two conjectures posed by Kiguradze that Kneser solutions are determined by their value in
a point and that blow-up solutions have prescribed asymptotics.
Dedicated to Professor Vladimir Maz'ya on the occasion of his 60th birthday.
The author was supported by the Swedish Natural Science Research Council (NFR) grant M-AA/MA 10879-304. 相似文献
20.
We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S n+1 satisfying Sf 4 f_3~2 ≤ 1/n S~3 , where S is the squared norm of the second fundamental form of M, and f_k =sum λ_i~k from i and λ_i (1 ≤ i ≤ n) are the principal curvatures of M. We prove that there exists a positive constant δ(n)(≥ n/2) depending only on n such that if n ≤ S ≤ n + δ(n), then S ≡ n, i.e., M is one of the Clifford torus S~k ((k/n)~1/2 ) ×S~... 相似文献