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Let \(w=(w_ 1,w_ 2,\dots,w_ n)\) be a permutation in the symmetric group \(S_ n\). An explicit combinatorial construction of the Schubert polynomial \({\mathcal S}_ w\) is given as a sum of weights \(x^ D\) of diagrams \(D\) (finite nonempty sets of lattice points \((i,j)\) in the positive quadrant), where the sum is over the set \(\Omega(w)\) of diagrams \(D\) which are obtainable from an initial diagram \(D(w)\) by repeated application of allowable \(B\)-moves. If we view a diagram as a set of checkers (or black stones for those who prefer) distributed over the positive quadrant, then a \(B\)-move is just a move of one checker from a position to an adjacent unoccupied position subject to a set of rules, thereby obtaining a new diagram. Since the classical Schur functions can be viewed as special cases of the Schubert polynomials for appropriate choices of the permutation \(w\), it has been hoped that there was a combinatorial algorithm for constructing Schubert polynomials which extends (or is at least in analogy to) the construction of the Schur function \(S_ \lambda(x_ 1,\dots,x_ r)\) as a sum of weights of column strict tableaux of shape \(\lambda\), where \(\lambda\) is a partition. The algorithm given here is not quite an extension of the usual construction for Schur functions, however a simple bijection between \(\Omega(w)\) and the set of column strict tableaux of shape \(\lambda\) is given in the particular case where the Schubert polynomial \({\mathcal S}_ w\) is the Schur function \(S_ \lambda(x_ 1,\dots,x_ r)\). A different algorithm for constructing the general Schubert polynomial \({\mathcal S}_ w\) was conjectured (and proved when \(w\) is a vexillary permutation) by \textit{A. Kohnert} [Weintrauben, Polynome, Tableaux, Bayreuther Math. Schr. 38, 1-97 (1991; Zbl 0755.05095)]. A sketch is given at the end of the paper here of how one would show the equivalence of the rule given here and that of Kohnert. Die vorliegende Arbeit beschäftigt sich mit einer neuen kombinatorischen Integerpretation von Polynomen, die sich im Zusammenhang mit Schubertpolynomen ergeben. Dies sind insbesondere Schurpolynome und Gaußpolynome. Bei den neuen kombinatorischen Objekten, den Weintrauben, handelt es sich um 0-1-Matrizen. Mittels einer sehr einfachen Operation kann aus einem Startobjekt \(w\) eine endliche Menge von Weintrauben erzeugt werden. Diese Menge \(S(w)\) kann dann als Polynom oder als Menge von Tableaux interpretiert werden. Diese Grundlagen werden in Kapitel I definiert.
Schubertpolynome sind Polynome in mehreren Variablen und wurden erstmals von Lascoux und Schützenberger in einer 1982 erschienenen Arbeit definiert. Sie werden durch Permutationen indiziert, und im Falle von Permutationen mit einem einzigen Abstieg sind es Schurpolynome. Auf diese Zusammenhänge wird im Kapitel II eingegangen. Da es für Schurpolynome die bekannte kombinatorische Interpretation als Summe von Tableaux mit gegebenem Umriß und gegebenem maximalen Eintrag gibt, stellt sich die Frage nach der Interpretation der Schubertpolynome. Diese Frage wurde von Lascoux und Schützenberger in einer Arbeit aus dem Jahr 1988 beantwortet. Ein Schubertpolynom ist demnach eine Vereinigung von Tableauxmengen \(T_ I\), wobei jede einzelne Menge eine Teilmenge einer Tableauxmenge eines Schurpolynoms ist und verschiedene Mengen aus Tabeleaux mit verschiedenem Umriß bestehen können.
Im Kapitel III werden spezielle Weintrauben \(w\) definiert, so daß man die Schurpolynome, Gaußpolynome und auch die Schiefschurpolynome als \(S(w)\) erhält. Hierbei wird \(S(w)\) wie oben erwähnt als Polynom interpretiert. Dies ermöglicht z.B. einen einfachen Algorithmus zur Erzeugung aller Schieftableaux. Ferner gelingt es auch, die Menge \(T_ I\) in der Form \(S(w)\) dazustellen. | 1 |
Let \(w=(w_ 1,w_ 2,\dots,w_ n)\) be a permutation in the symmetric group \(S_ n\). An explicit combinatorial construction of the Schubert polynomial \({\mathcal S}_ w\) is given as a sum of weights \(x^ D\) of diagrams \(D\) (finite nonempty sets of lattice points \((i,j)\) in the positive quadrant), where the sum is over the set \(\Omega(w)\) of diagrams \(D\) which are obtainable from an initial diagram \(D(w)\) by repeated application of allowable \(B\)-moves. If we view a diagram as a set of checkers (or black stones for those who prefer) distributed over the positive quadrant, then a \(B\)-move is just a move of one checker from a position to an adjacent unoccupied position subject to a set of rules, thereby obtaining a new diagram. Since the classical Schur functions can be viewed as special cases of the Schubert polynomials for appropriate choices of the permutation \(w\), it has been hoped that there was a combinatorial algorithm for constructing Schubert polynomials which extends (or is at least in analogy to) the construction of the Schur function \(S_ \lambda(x_ 1,\dots,x_ r)\) as a sum of weights of column strict tableaux of shape \(\lambda\), where \(\lambda\) is a partition. The algorithm given here is not quite an extension of the usual construction for Schur functions, however a simple bijection between \(\Omega(w)\) and the set of column strict tableaux of shape \(\lambda\) is given in the particular case where the Schubert polynomial \({\mathcal S}_ w\) is the Schur function \(S_ \lambda(x_ 1,\dots,x_ r)\). A different algorithm for constructing the general Schubert polynomial \({\mathcal S}_ w\) was conjectured (and proved when \(w\) is a vexillary permutation) by \textit{A. Kohnert} [Weintrauben, Polynome, Tableaux, Bayreuther Math. Schr. 38, 1-97 (1991; Zbl 0755.05095)]. A sketch is given at the end of the paper here of how one would show the equivalence of the rule given here and that of Kohnert. This paper considers a one-dimensional generalized Allen-Cahn equation of the form
\[
u_t = \varepsilon^2 (D(u)u_x)_x - f(u),
\]
where \(\varepsilon > 0\) is constant, \(D = D(u)\) is a positive, uniformly bounded below, diffusivity coefficient that depends on the phase field \(u\), and \(f(u)\) is a reaction function that can be derived from a double-well potential with minima at two pure phases \(u = \alpha\) and \(u = \beta\). It is shown that interface layers (namely, solutions that are equal to \(\alpha\) or \(\beta\) except at a finite number of thin transitions of width \(\varepsilon)\) persist for an exponentially long time proportional to \(\exp (C/\varepsilon)\), where \(C > 0\) is a constant. In other words, the emergence and persistence of metastable patterns for this class of equations is established. For that purpose, we prove energy bounds for a renormalized effective energy potential of Ginzburg-Landau type. Numerical simulations, which confirm the analytical results, are also provided. | 0 |
The main result of this paper is that if M is a compact Kähler manifold with positive first Chern class \(c_ 1(M)\) then an invariant \(\eta\) (M) may be defined such that \(\eta (M)<1\) implies the existence of a Kähler-Einstein metric on M. \(\eta\) (M) is obtained from the study of an inequality discussed by \textit{T. Aubin} [J. Funct. Anal. 57, 143-153 (1984; Zbl 0538.53063)]. On démontre l'existence d'une métrique d'Einstein-Kähler sur une variété Kählérienne compacte à première classe de Chern \(C_ 1\) positive. Il est bien connu que dans ce cas il existe des variétés Kählériennes compactes n'admettant pas d'une métrique d'Einstein- Kähler. La méthode utilisée est celui de continuité, qui permet de ramener le preuve à la démonstration d'une inégalité. Le résultat principal est le suivant; il existe une métrique d'Einstein- Kähler si \(C^ m_ 1<(m+1)^{2m}\quad(2m)^{-m}\) (\(m=\) dimension complexe de la variété). | 1 |
The main result of this paper is that if M is a compact Kähler manifold with positive first Chern class \(c_ 1(M)\) then an invariant \(\eta\) (M) may be defined such that \(\eta (M)<1\) implies the existence of a Kähler-Einstein metric on M. \(\eta\) (M) is obtained from the study of an inequality discussed by \textit{T. Aubin} [J. Funct. Anal. 57, 143-153 (1984; Zbl 0538.53063)]. The authors consider time-optimal dampening of oscillations of a mathematical pendulum by using as a damper a pendulum with variable length and a small mass. In the case of small oscillations a piecewise- constant law of varying the length of the dampening pendulum is determined from the maximum principle. A numerical example is given. | 0 |
It was shown by \textit{P. F. Smith} and the reviewer that a ring \(R\) is left noetherian if and only if there exists a cardinal \(c\) such that every direct sum of injective left \(R\)-modules is a direct sum of an injective module and a \(c\)-limited module [Commun. Algebra 18, No. 6, 1971-1988 (1990; Zbl 0713.16004)]. Motivated by this, the author obtains the following general result: Let \(P\) and \(Q\) be properties of left \(R\)- modules such that \(P\) is preserved by direct sums and \(Q\) is inherited by direct summands. If there is a cardinal \(c\) such that every left \(R\)- module with property \(P\) is the direct sum of a module with property \(Q\) and a \(c\)-limited module with essential socle, then every left \(R\)-module with property \(P\) has property \(Q\). Some new characterizations of semisimple rings, left perfect rings, left noetherian rings, von Neumann regular rings, QF-rings and IF-rings arise as corollaries. Let c be a cardinal number. A module M is said to be c-limited if every internal direct sum of nonzero submodules of M has at most c terms. Let R be a ring. It is shown that R is right Noetherian if and only if there is a cardinal number c such that every direct sum of injective right R- modules is the direct sum of an injective module and a c-limited module. Also the following are equivalent: (1) R is semi-simple Artinian; (2) there is a cardinal number c such that every right R-module is the direct sum of an injective module and a c-limited module; (3) as (2) but with ``projective'' in place of ``injective''. This leads to the following question: Let R be a ring such that every cyclic right R-module is the direct sum of an injective module and a Noetherian module; is R right Noetherian? It is shown that the answer is ``Yes'' if R is commutative or semi-prime or has only one minimal prime ideal or R/N has zero socle (where N is the prime radical of R), but an example is given to show that the answer can be ``No''. | 1 |
It was shown by \textit{P. F. Smith} and the reviewer that a ring \(R\) is left noetherian if and only if there exists a cardinal \(c\) such that every direct sum of injective left \(R\)-modules is a direct sum of an injective module and a \(c\)-limited module [Commun. Algebra 18, No. 6, 1971-1988 (1990; Zbl 0713.16004)]. Motivated by this, the author obtains the following general result: Let \(P\) and \(Q\) be properties of left \(R\)- modules such that \(P\) is preserved by direct sums and \(Q\) is inherited by direct summands. If there is a cardinal \(c\) such that every left \(R\)- module with property \(P\) is the direct sum of a module with property \(Q\) and a \(c\)-limited module with essential socle, then every left \(R\)-module with property \(P\) has property \(Q\). Some new characterizations of semisimple rings, left perfect rings, left noetherian rings, von Neumann regular rings, QF-rings and IF-rings arise as corollaries. We study the nonhomogeneous Dirichlet problem for first-order Hamilton-Jacobi equations associated with Tonelli Hamiltonians on a bounded domain \(\Omega\) of \(\mathbb{R}^n\) assuming the energy level to be supercritical. First, we show that the viscosity (weak KAM) solution of such a problem is Lipschitz continuous and locally semiconcave in \(\Omega\). Then, we analyze the singular set of a solution showing that singularities propagate along suitable curves, the so-called generalized characteristics, and that such curves stay singular unless they reach the boundary of \(\Omega\). Moreover, we prove that the latter is never the case for mechanical systems and that singular generalized characteristics converge to a critical point of the solution in finite or infinite time. Finally, under stronger assumptions for the domain and Dirichlet data, we are able to conclude that solutions are globally semiconcave and semiconvex near the boundary. | 0 |
For an algebra \textbf{A}, \(p_n({\mathbf A})\) denotes the number of essentially \(n\)-ary term functions over \textbf{A}. It was shown by \textit{G. Grätzer} and \textit{R. Padmanabhan} [Proc. Am. Math. Soc. 28, 75-80 (1971; Zbl 0215.34501)] that an algebra \textbf{A} is a nontrivial affine space over GF(3) if and only if \(p_n({\mathbf A})=(2^n-(-1)^n)/3\) for \(n=1,2,\dots\). The author of this paper gives various results where extra conditions on \textbf{A} enable a weakening of the conditions on \(p_n({\mathbf A})\). The particular result given in this note is that an idempotent algebra \((G,s)\) of type (3) is a nontrivial affine space over GF(3) if and only if \(p_4(G,s)=5\). For an algebra \( \mathfrak{A} = \langle A;F\rangle\) and for \(n\geq2\), let \(p_n(\mathfrak{A})\) denote the number of essentially \(n\)-ary polynomials of \(\mathfrak{A}\). \textit{J. Dudek} [Colloq. Math. 21, 169--177 (1970; Zbl 0216.08904)] has shown that if \(\mathfrak{A}\) is an idempotent and nonassociative groupoid then \(p_n(\mathfrak{A}) \geq n\) for all \(n>2\). In this paper this result is improved for the commutative case to show that for such groupoids \(\mathfrak{A}\),
\[
p_n(\mathfrak{A}) \geq \tfrac13({2^n}-{(-1)^n})
\]
for all \(n\geq 2\) (Theorem 1) and that this is the best possible result. Those groupoids for which this lower bound is attained are completely characterized. In fact, the relevant result proved below is much stronger (Theorem 3). From these and other known results it is deduced that the sequence \(\langle 0,0,1,3\rangle \) has the minimal extension property. | 1 |
For an algebra \textbf{A}, \(p_n({\mathbf A})\) denotes the number of essentially \(n\)-ary term functions over \textbf{A}. It was shown by \textit{G. Grätzer} and \textit{R. Padmanabhan} [Proc. Am. Math. Soc. 28, 75-80 (1971; Zbl 0215.34501)] that an algebra \textbf{A} is a nontrivial affine space over GF(3) if and only if \(p_n({\mathbf A})=(2^n-(-1)^n)/3\) for \(n=1,2,\dots\). The author of this paper gives various results where extra conditions on \textbf{A} enable a weakening of the conditions on \(p_n({\mathbf A})\). The particular result given in this note is that an idempotent algebra \((G,s)\) of type (3) is a nontrivial affine space over GF(3) if and only if \(p_4(G,s)=5\). Ueber einen Theil der vorliegenden Arbeit ist bereits im vorigen Jahresbericht (F. d. M. III. p. 486-487, JFM 03.0486.02) nach einem in den Comptes rendus enthaltenen Auszuge berichtet. Wir knüpfen an jenen Bericht, in dem die zu Grunde gelegten Voraussetzungen und die Methode der Entwickelung angegeben sind, an, indem wir nur bemerken, dass es dort p. 486 in der letzten Zeile heissen muss: auf einer Verticalen \(y=\) const., während \(x\) der horizontalen Länge des Kanals parallel ist. Bezeichnet \(H\) die constante Kanaltiefe, \(h\) die (sehr kleine) Erhebung der Welle über dem ursprünglichen Niveau, \(g\) die Constante der Schwerkraft, so führt jene Methode in erster Annäherung auf die Differentialgleichung:
\[
\frac{ \partial^2 h}{ \partial t^2} =gH\;\frac{ \partial^2 h}{ \partial x^2},
\]
während die weitere Annäherung ergiebt:
\[
(1) \quad \frac{ \partial ^2 h}{ \partial t^2} =gH \frac{ \partial^2 h}{ \partial x^2} +gH \frac{ \partial^2}{ \partial x^2} \left( \frac{ 3h^2}{ 2H} +\frac{ H^2}{3} \frac{ \partial^2 h}{ \partial x^2} \right).
\]
Hierzu kommt für die Fortpflanzungsgeschwindigkeit \(w\) die Gleichung:
\[
(2) \quad \frac{ \partial h}{ \partial t} +\frac{ \partial (hw)}{ \partial x} =0.
\]
Letztere Gleichung drückt aus, dass jedes Element der Welle beim Fortschreiten dasselbe Volumen behält. Aus diesen Gleichungen wird nun mit Vernachlässigung aller Glieder, die die vorliegende Näherung übersteigen, folgende Relation abgeleitet:
\[
(3) \quad h( w- \sqrt{gH} ) -\frac{ \sqrt{gH}}{2} \left( \frac{3h^2}{ 2H} +\frac{H^2}{3}\;\frac{ \partial ^2 h}{ \partial x^2} \right) =\chi (x +t\sqrt{ gH}),
\]
wo \(\chi\) eine willkürliche Function bezeichnet. Nimmt man nun in Bezug auf den Anfangszustand an, dass für \(t=0 h\) und seine Ableitung nach \(x\) verschwinden für alle positiven \(x\), so ist für alle Theilchen, die dem vorderen Ende der fortschreitenden Wellenerhebung nahe sind, \(\chi=0\), und aus der Gleichung (3) folgt für die Fortpflanzungsgeschwindigkeit eines Elements der Welle:
\[
(3a) \quad w^2 =g\; \left(H +\frac{ 3h}{2} +\frac{H^2}{ 3h}\;\frac{\partial^2h}{ \partial x^2} \right),
\]
so dass die Fortpflanzungsgeschwindigkeit auch von der Krümmung der freien Oberfläche abhängt. Es folgen nun noch einige Umformungen der Gleichung (2) für \(h\), ferner die Formeln, durch welche bei der hier durchgeführten Näherung die Geschwindigkeiten und der Druck bestimmt werden.
Der folgende Abschnitt behandelt die Bewegung des Schwerpunkts der ganzen Welle. Das Quadrat der Fortpflanzungsgeschwindigkeit dieses Punktes ist \(=g(H+ 3\eta)\), wo \(\eta\) die Höhe des Schwerpunktes über dem ursprünglichen Niveau ist; \(\eta\) ist constant, so dass der Schwerpunkt sich auf einer geraden Linie bewegt. - Von besonderem Interesse ist diejenige Welle, bei der alle Theilchen sich mit gleicher Geschwindigkeit fortpflanzen (so dass \(w\) constant ist), und die bei ihrer Fortpflanzung nahezu dieselbe Form behält (onde solitaire). Die freie Oberfläche dieser Welle wird bestimmt durch die Gleichung
\[
h= \frac{4h_1}{ 2+e^{ \scriptstyle \sqrt{ \frac{3h_1}{H^3}} (x-wt)} +e^{ \scriptstyle -\sqrt{ \frac{ 3h_1}{ H^3}} (x-wt)} },
\]
wenn \(w^2 =g (H+h_1)\) der constante Werth von \(w\) ist. Der Schwerpunkt dieser Welle liegt in \(\tfrac{1}{3}\) ihrer Höhe.
Moment der Instabilität nennt ferner der Verfasser folgendes Integral:
\[
M=\int^{\infty}_{x_0} \left[ \left(\frac{ \partial h}{ \partial x}\right)^2 -\frac{3h^3}{ H^3} \right] dx.
\]
Dasselbe ist ein Minimum für die onde solitaire. Für irgend eine andere Welle kann der Ueberschuss des wirklichen Werthes von \(M\) über den Minimalwerth betrachtet werden als Maass der Abweichung der Gestalt dieser Welle von der onde solitaire, sowie auch als Maass für die Deformation, welche die Welle bei ihrer Fortpflanzung erleidet. Weitere Discussionen und Erläuterungen der obigen Formeln bilden den Schluss der Arbeit. | 0 |
Let \(\| \cdot \| \) be a norm on \(\mathbb R^n\). Consider the following unconditional norm on \(\mathbb R^n\)
\[
| | | (x_1, \dots, x_n)| | | := \text{Ave} \biggl\| \sum_{i=1}^n \varepsilon_i x_i e_i\biggr\|,
\]
where the average is taken over all \(2^n\) choices of signs \(\varepsilon_i = \pm 1\). \textit{J.~Bourgain, J.~Lindenstrauss} and \textit{V. D.~Milman} [Lect. Notes Math. 1317, 44--66 (1988; Zbl 0645.52001)] showed that there exists an absolute constant \(C>1\) such that it is enough to take the average over \(N:= C n\) (random) choices of vectors \((\varepsilon _1, \dots, \varepsilon _n)\) in order to obtain a norm isomorphic to \(| | | \cdot| | | \). In the present paper, the authors improve the result to the case of an arbitrary \(N>n\) with constants of isomorphism depending only on \((N-n)/n\). [For the entire collection see Zbl 0638.00019.]
By using the empirical distribution method from probability theory and inequalities from Banach space theory, the authors establish some very interesting results on Minkowski sums of convex bodies. In various different ways, these results say that the effect of adding many convex bodies can approximately be reached by adding only relatively few bodies. This is made precise by a series of rather sharp estimates, of which we give two examples. Let \(\{K_ i\}^ m_{i=1}\) be (centrally) symmetric convex bodies in \({\mathbb{R}}^ n\) and let \(K=\Sigma^ m_{i=1}K_ i\). Then for every \(0<\epsilon <1/2\) there is a subset \(\{i_ j\}^ N_{j=1}\) of \(\{\) 1,..., \(m\}\) with \(N\leq cn^ 2\epsilon^{-2}| \log \epsilon |\) and scalars \(\{\lambda_ j\}^ N_{j=1}\) so that
\[
(1-\epsilon)K \subset \sum^{N}_{j=1}\lambda_ jK_{i_ j} \subset (1+\epsilon)K.
\]
(c denotes an absolute constant). Let K be a symmetric convex body in \({\mathbb{R}}^ n\) and let \(\epsilon >0\). If \(n>n_ 0(\epsilon)\) and if we perform \(N=cn\log n+c(\epsilon)n\) random Minkowski symmetrizations on K we obtain with probability \(1-\exp (-\tilde c(\epsilon)n)\) a body \(\tilde K\) which satisfies
\[
(1-\epsilon)B \subset \tilde K \subset (1+\epsilon)B
\]
for a suitable ball \(B=B(K)\). Here the random Minkowski symmetrizations are performed consecutively with respect to hyperplanes through 0 whose normal vectors are chosen independently and uniformly. | 1 |
Let \(\| \cdot \| \) be a norm on \(\mathbb R^n\). Consider the following unconditional norm on \(\mathbb R^n\)
\[
| | | (x_1, \dots, x_n)| | | := \text{Ave} \biggl\| \sum_{i=1}^n \varepsilon_i x_i e_i\biggr\|,
\]
where the average is taken over all \(2^n\) choices of signs \(\varepsilon_i = \pm 1\). \textit{J.~Bourgain, J.~Lindenstrauss} and \textit{V. D.~Milman} [Lect. Notes Math. 1317, 44--66 (1988; Zbl 0645.52001)] showed that there exists an absolute constant \(C>1\) such that it is enough to take the average over \(N:= C n\) (random) choices of vectors \((\varepsilon _1, \dots, \varepsilon _n)\) in order to obtain a norm isomorphic to \(| | | \cdot| | | \). In the present paper, the authors improve the result to the case of an arbitrary \(N>n\) with constants of isomorphism depending only on \((N-n)/n\). It is well-known that
\[
\liminf_{t\to\infty} \int^t_{t-\tau} p(s)ds>1/e
\]
implies that all solutions of
\[
y'(t)+ p(t)y (t-\tau) =0,\;t\geq t_0 \tag{*}
\]
oscillate, where \(\tau>0\) and \(p\in C ([t_0, \infty), [0,\infty))\). \textit{B. Li} has obtained [J. Math. Anal. Appl. 192, No. 1, 312-321 (1995; Zbl 0829.34060)] the following sharper sufficient conditions for oscillation of all solutions of (*): There exists a \(\widetilde t>t_0 +\tau\) such that
\[
\int^t_{t-\tau} p(s)ds \geq 1/e, \quad t\geq \widetilde t \tag{**}
\]
and
\[
\int^\infty_{t_0+ \tau} p(t)\left[\exp \left(\int^t_{t-\tau} p(s)ds-1/e \right)-1 \right]dt =\infty.
\]
The conditions (**) are improved further. If there exist a \(t_1>t_0 +\tau\) and a positive integer \(n\) such that
\[
p_n(t)\geq e^{-n}, \quad \overline p_n(t) \geq e^{-n},\;t\geq t_1
\]
and
\[
\int^\infty_{t_0+n \tau} p(t)\biggl[\exp \bigl(e^{n-1} p_n(t)- e^{-1} \bigr)-1 \biggr]dt=\infty,
\]
then all solutions of (*) oscillate, where
\[
\begin{aligned} p_1(t) & =\int^t_{t-\tau} p(s)ds,\;t\geq t_0 +\tau, \\ p_{k+1}(t) & =\int^t_{t-\tau} p(s)p_k(s)ds,\;t\geq t_0+ (k+1)\tau \\ \overline p_1 (t) & = \int^{t +\tau}_t p(s)ds,\;t\geq t_0,\\ \overline p_{k+1} (t) & =\int^{t+ \tau}_t p(s)\overline p_k(s)ds,\;t\geq t_0, \;k=1,2, \dots, \end{aligned}
\]
An example is given to illustrate the result. | 0 |
The paper discusses the properties of the indices of the Cartesian product \(A\times B\) such as \(\dim_{H}\), \(\dim_{K}\), \(\dim_{L}\), \(\triangle ^{*}\), and \(\dim_{M}\), where \(A\) and \(B\) are subsets of \(\mathbb{Z}^{d}\). In [Proc. Lond. Math. Soc., III. Ser. 64, 125-152 (1992; Zbl 0753.28006)] \textit{M. T. Barlow} and \textit{S. J. Taylor} showed that for any \(A\subseteq \mathbb{Z}^{d_1}\) and \(B\subseteq \mathbb{Z}^{d_2}\), the following are true: (1) \(\dim_{K}(A\times B)\geq \dim_{K}(A)+\dim_{K}(B)\); (2) \(\dim_{L}(A\times B)\geq \dim_{L}(A)+\dim_{K}(B)\); (3) \(\dim_{H}(A\times B)\geq \dim_{H}(A)+\dim_{K}(B)\); (4) \(\dim_{H}(A\times B)\leq \dim_{H}(A)+\triangle ^{*}(B)\); (5) \(\triangle ^{*}(A\times B)\geq \dim_{K}(A)+\triangle ^{*}(B)\); (6) \(\triangle ^{*}(A\times B)\leq \triangle ^{*}(A)+\triangle ^{*}(B)\).
In this paper the author shows that (a) \(\triangle (A\times B)\geq \triangle (A)+\dim_{K}(B)\); (b) \(\triangle (A\times B)\leq \triangle (A)+\triangle ^{*}(B)\); (c) \(\dim_{K}(A\times B)\leq \dim_{K}(A)+\triangle ^{*}(B)\); (d) \(\dim_{L}(A\times B)\leq \dim_{L}(A)+\triangle ^{*}(B)\). This paper investigates the possible definitions of dimension for sets in a lattice such as \(\mathbb{Z}^ d\). This seems to be necessary since many models in statistial physics are based on \(\mathbb{Z}^ d\) (for example the DLA model and the percolation model) rather than on \(\mathbb{R}^ d\). Thus the authors introduce discrete Hausdorff and discrete packing dimension, which are devoted to the macroscopic behaviour of a subset. Then a subset of \(\mathbb{Z}^ d\) is said to be a fractal if discrete Hausdorff and packing dimension agree. They show how to produce fractals of non-integer dimension and develop the discrete version of the Frostman lemma to calculate such discrete dimensions. This includes some discrete potential theory. The main result is that the range of a strictly stable random walk in \(\mathbb{Z}^ d\) is almost surely a fractal. The paper should be useful to give rigorous proofs for dimension results in statistical physics. | 1 |
The paper discusses the properties of the indices of the Cartesian product \(A\times B\) such as \(\dim_{H}\), \(\dim_{K}\), \(\dim_{L}\), \(\triangle ^{*}\), and \(\dim_{M}\), where \(A\) and \(B\) are subsets of \(\mathbb{Z}^{d}\). In [Proc. Lond. Math. Soc., III. Ser. 64, 125-152 (1992; Zbl 0753.28006)] \textit{M. T. Barlow} and \textit{S. J. Taylor} showed that for any \(A\subseteq \mathbb{Z}^{d_1}\) and \(B\subseteq \mathbb{Z}^{d_2}\), the following are true: (1) \(\dim_{K}(A\times B)\geq \dim_{K}(A)+\dim_{K}(B)\); (2) \(\dim_{L}(A\times B)\geq \dim_{L}(A)+\dim_{K}(B)\); (3) \(\dim_{H}(A\times B)\geq \dim_{H}(A)+\dim_{K}(B)\); (4) \(\dim_{H}(A\times B)\leq \dim_{H}(A)+\triangle ^{*}(B)\); (5) \(\triangle ^{*}(A\times B)\geq \dim_{K}(A)+\triangle ^{*}(B)\); (6) \(\triangle ^{*}(A\times B)\leq \triangle ^{*}(A)+\triangle ^{*}(B)\).
In this paper the author shows that (a) \(\triangle (A\times B)\geq \triangle (A)+\dim_{K}(B)\); (b) \(\triangle (A\times B)\leq \triangle (A)+\triangle ^{*}(B)\); (c) \(\dim_{K}(A\times B)\leq \dim_{K}(A)+\triangle ^{*}(B)\); (d) \(\dim_{L}(A\times B)\leq \dim_{L}(A)+\triangle ^{*}(B)\). The author has shown that there exists a regular \(T_ 1\) space X and a metric space Y such that \(\tau (X\times Y)\neq \tau (X)\times Y,\) where \(\tau\) is the Tychonoff functor introduced by \textit{K. Morita} [Fundamenta Math. 87, 31-52 (1975; Zbl 0336.55003)]. This answers in negative the question posed by the author in his earlier paper (Problem 5.3) [Math. Jap. 29, 847-858 (1984; Zbl 0576.54029)]. | 0 |
From the Abstract: ``This paper addresses the long-time behaviour of gradient flows of nonconvex functionals in Hilbert spaces. Exploiting the notion of generalized semiflows by \textit{J. M. Ball} [J. Nonlinear Sci. 7, No.~5, 475--502 (1997); erratum ibid. 8, 233 (1998; Zbl 0903.58020)], we provide some sufficient conditions for the existence of a global attractor. The abstract results are applied to various classes of nonconvex evolution problems. In particular, we discuss the long-time behaviour of solutions of quasistationary phase field models and prove the existence of a global attractor.'' A class of semiflows (generalized semiflows) with possibly nonunique solutions is considered. A generalized semiflow is defined to be a family of maps \(\varphi : [0,\infty)\to X\), where \(X\) is a metric space, satisfying axioms relating to existence, time translation, concatenation, and upper-semicontinuity with respect to initial data. It is shown that, under a mild technical hypothesis, for generalized semiflows strong measurability of solutions with respect to time implies their continuity on \((0,\infty)\). A generalized semiflow is shown to have a global attractor if and only if it is point dissipative and asymptotically compact. This result generalizes those for semiflows of J. K. Hale and O. A. Ladyzhenskaya. The structure of the global attractor in the presence of a Lyapunov function and its connectedness and stability properties are also studied. In particular, two examples (one finite- and the other infinite-dimensional) are given in which the global attractor is a single point but is not Lyapunov stable. Conditions under which the global attractor of a generalized semiflow is stable are also obtained. The theory of generalized semiflows is applied to the study of the 3D incompressible Navier-Stokes equations. It is shown that weak solutions satisfying an energy inequality form a generalized semiflow (in the phase space \(H\) consisting of \(L^2\) vector-fields with zero divergence) if and only if all weak solutions are continuous from \((0,\infty)\) to \(L^2\). Under the (unproved) hypothesis that all weak solutions are continuous from \((0,\infty)\) to \(L^2\), the existence of a global attractor is established. | 1 |
From the Abstract: ``This paper addresses the long-time behaviour of gradient flows of nonconvex functionals in Hilbert spaces. Exploiting the notion of generalized semiflows by \textit{J. M. Ball} [J. Nonlinear Sci. 7, No.~5, 475--502 (1997); erratum ibid. 8, 233 (1998; Zbl 0903.58020)], we provide some sufficient conditions for the existence of a global attractor. The abstract results are applied to various classes of nonconvex evolution problems. In particular, we discuss the long-time behaviour of solutions of quasistationary phase field models and prove the existence of a global attractor.'' What is the relationship between the degree of learning difficulty of a Boolean concept (i.e., a category defined by logical rules expressed in terms of Boolean operators) and the complexity of its logical description? Feldman [(2000). Minimization of Boolean complexity in human concept learning. Nature, 407(October), 630--633] investigated this question experimentally by defining the complexity of a Boolean formula (that logically describes a concept) as the length of the shortest formula logically equivalent to it. Using this measure as the independent variable in his experiment, he concludes that in general, the subjective difficulty of learning a Boolean concept is well predicted by Boolean complexity. Moreover, he claims that one of the landmark results and benchmarks in the human concept learning literature, the Shepard, Hovland, and Jenkins learning difficulty ordering, is precisely predicted by this hypothesis. However, in what follows, we introduce a heuristic procedure for reducing Boolean formulae, based in part on the well-established minimization technique from Boolean algebra known as the Quine--McCluskey (QM) method, which when applied to the SHJ Boolean concept types reveals that some of their complexity values are notably different from the approximate values obtained by Feldman. Furthermore, using the complexity values for these simpler expressions fails to predict the correct empirical difficulty ordering of the SHJ concept types. Motivated by these findings, this note includes a brief tutorial on the QM method and concludes with a brief discussion on some of the challenges facing the complexity hypothesis. | 0 |
The initial value problem defined by a system of nonlinear ordinary differential equations is reformulated as a fixed point problem of a certain operator \(T\) on a Banach space which is subsequently solved by applying an asynchronous iteration associated with \(T\) (the so-called asynchronous multi-splitting waveform relaxation). This technique generalizes the treatment introduced by \textit{A. Frommer} and \textit{B. Pohl} [Numer. Linear Algebra Appl. 2, No.~4, 335--346 (1995; Zbl 0831.65031)] for linear ordinary differential equations. The paper under review, based largely on previous results obtained by the first author, establishes the superlinear convergence of the algorithm. For certain multisplitting iterative methods to solve linear systems with an \(M\)-matrix it is shown that overlapping block iterations yield faster convergence than in the case of nonoverlapping blocks. This result is applied to variants of multisplitting waveform relaxation algorithms for solving discretized differential equations with initial conditions. Numerical experiments with different overlaps are carried out. | 1 |
The initial value problem defined by a system of nonlinear ordinary differential equations is reformulated as a fixed point problem of a certain operator \(T\) on a Banach space which is subsequently solved by applying an asynchronous iteration associated with \(T\) (the so-called asynchronous multi-splitting waveform relaxation). This technique generalizes the treatment introduced by \textit{A. Frommer} and \textit{B. Pohl} [Numer. Linear Algebra Appl. 2, No.~4, 335--346 (1995; Zbl 0831.65031)] for linear ordinary differential equations. The paper under review, based largely on previous results obtained by the first author, establishes the superlinear convergence of the algorithm. This paper reviews the state of the art in the theory of multisets, i.e., mathematical models of sets with repetitions (duplicates or copies of their elements). The corresponding bibliography is categorized as follows: the general theory of multisets, reviews, and application of multisets, in particular, in computer science. | 0 |
In this paper, the authors continue the study of Frobenius and separable cowreaths began in their previous paper [J. Noncommut. Geom. 9, 707--774 (2015; Zbl 1347.16035)], in the context of so-called \textit{Galois cowreaths}.
Let \(\mathcal C\) be a monoidal category with (co)equalizers (all of whose objects are assumed to be coflat and robust), and let \((A, X)\) be a Galois cowreath in \(\mathcal C\).
One of the main results of the paper establishes a one-to-one correspondence between Frobenius systems of the algebra extension \(A^{\mathrm{co}(X)} \to A\) in \(\mathcal C\) and Frobenius systems of a certain coalgebra \((X, \psi)\) associated to \((A, X)\). As a consequence of this it is shown that the extension \(A^{\mathrm{co}(X)} \to A\) is Frobenius if and only if the cowreath \((A, X)\) is Frobenius. Necessary and sufficient conditions
for a Frobenius pre-Galois cowreath to be Galois are also given.
Other results of the paper concern the notion of separability. In contrast with the previously mentioned result, the separability of the extension
\(A^{\mathrm{co}(X)} \to A\) is not equivalent to the separability of the cowreath \((A, X)\). The authors give here some necessary and sufficient conditions for the extension \(A^{\mathrm{co}(X)} \to A\) to be separable. They also show that a Frobenius Galois cowreath is separable if and only if it admits a total integral. We characterize Frobenius and separable monoidal algebra extensions \(i\colon R\to S\) in terms given by \(R\) and \(S\). For instance, under some conditions, we show that the extension is Frobenius, respectively separable, if and only if \(S\) is a Frobenius, respectively separable, algebra in the category of bimodules over \(R\). In the case when \(R\) is separable we show that the extension is separable if and only if \(S\) is a separable algebra. Similarly, in the case when \(R\) is Frobenius and separable in a sovereign monoidal category we show that the extension is Frobenius if and only if \(S\) is a Frobenius algebra and the restriction at \(R\) of its Nakayama automorphism is equal to the Nakayama automorphism of \(R\). As applications, we obtain several characterizations for an algebra extension associated to a wreath to be Frobenius, respectively separable. | 1 |
In this paper, the authors continue the study of Frobenius and separable cowreaths began in their previous paper [J. Noncommut. Geom. 9, 707--774 (2015; Zbl 1347.16035)], in the context of so-called \textit{Galois cowreaths}.
Let \(\mathcal C\) be a monoidal category with (co)equalizers (all of whose objects are assumed to be coflat and robust), and let \((A, X)\) be a Galois cowreath in \(\mathcal C\).
One of the main results of the paper establishes a one-to-one correspondence between Frobenius systems of the algebra extension \(A^{\mathrm{co}(X)} \to A\) in \(\mathcal C\) and Frobenius systems of a certain coalgebra \((X, \psi)\) associated to \((A, X)\). As a consequence of this it is shown that the extension \(A^{\mathrm{co}(X)} \to A\) is Frobenius if and only if the cowreath \((A, X)\) is Frobenius. Necessary and sufficient conditions
for a Frobenius pre-Galois cowreath to be Galois are also given.
Other results of the paper concern the notion of separability. In contrast with the previously mentioned result, the separability of the extension
\(A^{\mathrm{co}(X)} \to A\) is not equivalent to the separability of the cowreath \((A, X)\). The authors give here some necessary and sufficient conditions for the extension \(A^{\mathrm{co}(X)} \to A\) to be separable. They also show that a Frobenius Galois cowreath is separable if and only if it admits a total integral. Current and upcoming use of mobile devices and services allows information to be provided anytime and anywhere while inherent constraints on mobile devices caused difficulties on mobile Internet access. In this paper, we first explore impact of human preferences on user interface for mobile Internet access with user preferences survey. It was found that commonly used features and access methodologies are often not the human preferred ones and vice versa. It was found that hierarchical document browsing interface is much preferred by human while most commonly used interfaces are list-based. Such a gap has probably resulted in difficulties on mobile Internet access. Thus, we further proposed and explored the feasibility on mobile web document access via concept hierarchies (i.e. hierarchical interface or HI). Our initial results showed that an unconventional combination of term frequency and inverse document frequency yielded similar performance (i.e. 71\% ideal parent -child relationship) to previous work and the use of terms in titles achieved better performance than previous work (i.e. 82\% ideal parent -child relationship). Our initial result of building concept hierarchies after clustering compared to that without is encouraging (c.f. 82\% ideal parent -child relationship and 67\% ideal parent -child relationship). We believe that HI can be enhanced to a level for commercial deployment for mobile Internet access. | 0 |
The classical formula that represents all scattering matrices as a linear fractional transform of a Schur function is derived in a general setting of Pontryagin spaces. This formula is used for very general forms of classical interpolation problems of Nevanlinna-Pick type. This relates to the explicit formula for the \(\mathfrak{L}\)-resolvent of an isometric operator on such spaces. Therefore, boundary operator techniques are used as introduced in [\textit{T. Ya. Azizov} and \textit{I. S. Iokhvidov}, Foundations of the theory of linear operators in spaces with indefinite metric. (Osnovy teorii linejnykh operatorov v prostranstvakh s indefinitnoj metrikoj). Moskva: ''Nauka''. Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury (1986; Zbl 0607.47031)]. The authors present a detailed exposition of the up-to-date theory of linear operators on a Hilbert space \({\mathcal H}\) with indefinite inner product [x,y], a large part of which has been developed by Soviet mathematicians. When the indefinite inner product is generated by a non- trivial selfadjoint involution J, that is, \([x,y]=(Jx,y)\), the space is called a Krein space or a J-space. Let \({\mathcal H}_{\pm}=P_{\pm}({\mathcal H})\) where \(P_{\pm}=(I\pm J)/2\) are the canonical projections. When dim(\({\mathcal H}_+)<\infty\) or dim(\({\mathcal H}_-)<\infty\), the space is called a Pontrjagin space.
The book consists of five chapters. The first chapter is devoted to geometry of a Krein space. A subspace \({\mathcal L}\) is called nonnegative if [x,y]\(\geq 0\) for all x in \({\mathcal L}\). Nonpositivity and neutrality are correspondingly. Central concern here is the relationship between a nonnegative subspace and a contraction K from a subspace of \({\mathcal H}_+\) to \({\mathcal H}_-\); more precisely \({\mathcal L}=\{x\oplus Kx:\) \(x\in {\mathcal D}(K)\}\). This contraction is called the angular of \({\mathcal L}\). Thus a problem of extension of a nonnegative subspace to a maximal one is reduced to the problem of contractive extension of the angular operator to the whole \({\mathcal H}_+.\)
Fundamental classes of linear operators are inroduced in the second chapter. A linear operator T on a Krein space is called plus-operator if [Tx,Tx]\(\geq 0\) whenever [x,x]\(\geq 0\). For such an operator there exists \(\mu\geq 0\) such that [Tx,Tx]\(\geq \mu [x,x]\) for all x. If \(\mu >0\), T is a scalar multiple of a J-noncontractive operator S, i.e. [Sx,Sx]\(\geq [x,x].\)
The notions of J-nonexpansiveness, J-isometry and J-unitarity are also introduced. The J-adjoint of an operator is naturally defined. Even if an operator is J-noncontractive, its J-adjoint is not always so. If this is the case, the operator is called J-bi-noncontractive. It is shown that a J-noncontractive operator is J-bi-noncontractive if and only if the image of a maximal nonnegative subspace is again maximal nonnegative. A problem of extension of a J-nonnegative operator, i.e. [Tx,x]\(\geq 0\), is discussed by appealing to the theory of extension of positive symmetric operators.
In the third chapter the authors pose several questions about existence of a maximal semi-definite invariant subspace for an operator (or a family of operators) of suitable classes. An operator T (or a family of operators) is said to have \(\Phi_+\) (resp. \(\Phi_-)\) if it has a (common) invariant maximal nonnegative (resp. nonpositive) subspace \({\mathcal L}_+\) (resp. \({\mathcal L}_-)\). It has \(\Phi\) when it has both \(\Phi_+\) and \(\Phi_-\). When \({\mathcal L}_+\) and \({\mathcal L}_-\) are so chosen to be J-orthogonal it is said to have \(\Phi^{[\bot]}\). Further it is said to have \({\tilde \Phi}_+\) if every completely invariant nonnegative subspace is contained in an invariant maximal nonnegative subspace. \({\tilde \Phi}_-\), \({\tilde \Phi}\) and \({\tilde \Phi}^{[\bot]}\) are defined correspondingly. Main concern here is to obtain sufficient conditions for those properties \(\Phi\) 's. The following are among principal theorems.
A J-noncontractive operator T has \(\Phi_+\) if there exists a J-unitary operator V such such \(P_+(V^{-1}TV)P_-\) is compact.
A commutative group of J-unitary operators has \({\tilde \Phi}^{[\bot]}\) if it commutes with an operator \(V_ 0\) with \(\Phi\) such that every invariant nonnegative subspace of \(V_ 0\) becomes a J-orthogonal direct sum of a neutral subspace of finite dimension and a uniformly positive subspace.
The fourth chapter is devoted to spectral (i.e. integral) representation of J-selfadjoint operators. Though the theory can be extended to so- called definitable operators, the discussion is confined to the case of J-nonnegative operators. As an application, a J-polar decomposition of a J-bi-nonexpansive operator is given. Under suitable conditions a problem of completeness of eigen- and root- vectors of a J-dissipative operator is discussed.
In the final chapter the authors treat extension problems of J-isometric and J-symmetric operators. Here in place of the Cayley-Neumann transform the Potapov-Ginzburg transform is used to establish an indefinite analogue of an extensibility criterion. The lst section is devoted to a J-unitary dilation of a general operator and to a representation of any analytic operator-function as the transfer function of an operator quintet. The following are among main theorems.
For any bounded linear operator T on a Krein space (\({\mathcal H}_ 1,J_ 1)\) there exists a Krein space (\({\mathcal H}_ 2,J_ 2)\) and \(J_ 1\oplus J_ 2\)-unitary operator \(T=\left[ \begin{matrix} T_{11}&T_{12}\\ T_{21}& T_{22}\end{matrix} \right]\) on \({\mathcal H}_ 1\oplus {\mathcal H}_ 2\) such that \(T_{11}=T\) and \(T_{12}T^ n_{22}T_{21}=0\) for \(n=0,1,....\)
For any (\({\mathcal H}_ 1,J_ 1)\)-operator valued function T(\(\mu)\), analytic in the neighborhood of the origin, there exists a Krein space (\({\mathcal H}_ 2,J_ 2)\) and a \(J_ 1\oplus J_ 2\)-unitary operator \(T=\left[ \begin{matrix} T_{11}&T_{12}\\ T_{21}&T_{22}\end{matrix} \right]\) such that \(T(\mu)=T_{11}+T_{12}(I_ 2-\mu T_{22})^{- 1}T_{21}.\)
The book is clearly written and each section is supplied with a number of exercises. The only regrettable defect is the omission of an index.
This book by the experts can be recommended to everyone interested in the theory of spaces with indefinite metric. | 1 |
The classical formula that represents all scattering matrices as a linear fractional transform of a Schur function is derived in a general setting of Pontryagin spaces. This formula is used for very general forms of classical interpolation problems of Nevanlinna-Pick type. This relates to the explicit formula for the \(\mathfrak{L}\)-resolvent of an isometric operator on such spaces. Therefore, boundary operator techniques are used as introduced in [\textit{T. Ya. Azizov} and \textit{I. S. Iokhvidov}, Foundations of the theory of linear operators in spaces with indefinite metric. (Osnovy teorii linejnykh operatorov v prostranstvakh s indefinitnoj metrikoj). Moskva: ''Nauka''. Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury (1986; Zbl 0607.47031)]. We propose a new method to quantize gauge theories formulated on a canonical noncommutative spacetime with fields and gauge transformations taken in the enveloping algebra. We show that the theory is renormalizable at one loop and compute the beta function and show that the spin dependent contribution to the anomalous magnetic moment of the fermion at one loop has the same value as in the commutative quantum electrodynamics case. | 0 |
The author constructs a compactification for the relative degree-\(d\) Picard variety associated to a family of (proper) stable curves. The problem arises because the Picard functor neither is proper nor separated when fibers of \(X/S\) are not smooth. For instance one can find families of invertible sheaves specializing to sheaves not locally free when the central fiber has nodes. The author uses a new functor to solve this problem. One crucial idea is that in this new functor such families specialize to invertible sheaves on the curve resulting from replacing the node (in the central fiber) by \(\mathbb{P}^1\). [The same idea was used by \textit{D. Gieseker} for rank-2 vector bundles on nodal curves; cf. J. Differ. Geom. 19, 173-206 (1984; Zbl 0557.14008).]The proof relies on GIT (=``geometric invariant theory'') theory. In the present paper the author has two aims: 1. He develops a method for investigating the topology of the smooth projective variety \(U_ Y\) of isomorphism classes of stable bundles of rank two and degree d (d is odd) on a smooth projective curve of genus \(g\geq 2\); 2. As an application of the above method he proves the following conjecture of Newstead and Ramanan: ''The k-th Chern class of the tangent bundle of \(U_ Y\) is zero in the De Rham cohomology of \(U_ Y\) if \(k>2g-2\) (the ground field is the field of complex numbers).'' | 1 |
The author constructs a compactification for the relative degree-\(d\) Picard variety associated to a family of (proper) stable curves. The problem arises because the Picard functor neither is proper nor separated when fibers of \(X/S\) are not smooth. For instance one can find families of invertible sheaves specializing to sheaves not locally free when the central fiber has nodes. The author uses a new functor to solve this problem. One crucial idea is that in this new functor such families specialize to invertible sheaves on the curve resulting from replacing the node (in the central fiber) by \(\mathbb{P}^1\). [The same idea was used by \textit{D. Gieseker} for rank-2 vector bundles on nodal curves; cf. J. Differ. Geom. 19, 173-206 (1984; Zbl 0557.14008).]The proof relies on GIT (=``geometric invariant theory'') theory. The author considers locally \(m\)-convex algebras provided with \(C^*\)-seminorms and extends to this case some results known for usual \(C^*\)-algebras. In particular, she proves that every primitive ideal in such an algebra is closed and that the kernel of every continuous topologically irreducible representation is a closed primitive ideal. | 0 |
In the first part of the paper the author proves the boundedness of the classical pseudo-differential operators \(P=p(x,D)\) with \(p(x,\xi)\in S^m_{1,0}\), on Hölder-Zygmund spaces, namely \(P:\Lambda^s (\mathbb{R}^n)\to \Lambda^{s-m}(\mathbb{R}^n)\). With respect to other proofs [cf. \textit{M. E. Taylor}, Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials, Math. Surveys and Monographs 81 (2000; Zbl 0963.35211)], here explicit estimates of the norms of the operators are given via the seminorms of the symbols. This allows interesting applications to Fredholm property in \(\mathbb{R}^n\) and description of essential spectra of pseudodifferential operators on Hölder-Zygmund spaces. In the last part of the paper the author considers similar symbols which have an analytic extension with respect to the variable \(\xi\) in a tube domain \(\mathbb{R}^n+iA\). For them, the Fredholm property is proved in Hölder-Zygmund spaces with exponential weight. The book is devoted to develop related tools in the analysis of partial differential equations (PDEs):
1) Pseudodifferential operators with mildly regular symbols (estimates in various function spaces, coefficients in Sobolev spaces, commutator estimates).
2) Paradifferential operators and nonlinear estimates (product estimates, the use of maximal functions, estimates for compositions).
3) Applications to PDE (elliptic regularity, first-order operators, Dirichlet problem, parametrix estimates, Euler flows in rough domains, semilinear wave equations, divcurl estimates, propagation of singularities, and others).
4) Layer potentials on Lipschitz surfaces (Cauchy kernel, method of rotation and extensions, boundary integral operators, Dirichlet problem on Lipschitz domains, Koebe-Bieberbach distortion).
The book covers a lot of interesting and important material, mainly from harmonic analysis, the theory of function spaces and of course PDE-theory. It is a rather technical but very valuable book, in a sense it may serve as ``tool box'' for analysis working in the field(s). | 1 |
In the first part of the paper the author proves the boundedness of the classical pseudo-differential operators \(P=p(x,D)\) with \(p(x,\xi)\in S^m_{1,0}\), on Hölder-Zygmund spaces, namely \(P:\Lambda^s (\mathbb{R}^n)\to \Lambda^{s-m}(\mathbb{R}^n)\). With respect to other proofs [cf. \textit{M. E. Taylor}, Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials, Math. Surveys and Monographs 81 (2000; Zbl 0963.35211)], here explicit estimates of the norms of the operators are given via the seminorms of the symbols. This allows interesting applications to Fredholm property in \(\mathbb{R}^n\) and description of essential spectra of pseudodifferential operators on Hölder-Zygmund spaces. In the last part of the paper the author considers similar symbols which have an analytic extension with respect to the variable \(\xi\) in a tube domain \(\mathbb{R}^n+iA\). For them, the Fredholm property is proved in Hölder-Zygmund spaces with exponential weight. The author has proved some fixed point theorems on expansion mappings in 2-metric spaces which are straight-forward extensions of the corresponding results in metric spaces. As a sample, the following theorem may be cited from the paper. Theorem: Let S and T be two surjective orbitally continuous mappings of a complete 2-metric space X. If there exists a real number \(q>1\) such that \(d(Sx,Ty,a)\geq q \min \{d(x,Sx,a),d(y,Ty,a),d(x,y,a)\}\) for each x,y,a\(\in X\), then S and T have a common fixed point. | 0 |
Octahedral mechanisms consisting of rigid rods in the edges of an octahedron connected by spherical joints at the vertices of the octahedron were subject to intense studies by Bricard, Blaschke, Wunderlich and others. Recently these mechanisms attracted again attention in the study of Stewart-Gough parallel platform mechanisms [see \textit{A. Karger} and \textit{M. Husty}, Comput.-Aided Des. 30, No. 3, 205--215 (1998; Zbl 0908.68187)]. Because of the number of constraints these mechanisms are in general rigid. Bricard, Blaschke and recently Husty and Karger [loc. cit.] were interested in special geometries that allow finite motion of the mechanism. By extending the motion group to equiform motions every such mechanism has at least one degree of freedom. In the paper the authors study the motion related to an octahedral mechanism in the group of equiform motions and give a local parametrization of the motion. The `original' Stewart-Gough (S-G) platform is a parallel 6-6 mechanism with both bases isosceles triangles cut at vertices. All six points of the upper base are connected by telescopic legs with six points of the lower base by spherical joints. If the length of all six legs remains constant then the S-G platform is stiff (a structure), i.e. it has no movability in general. Exceptions from this rule are called self motions of the S-G platform and they can appear only in very special circumstances. They are known as Borel-Bricard motions, because they have been studied by Borel and Bricard at the beginning of this century [\textit{E. Borel}, Mémoire sur les déplacements à trajectories sphériques. Mémoires présentés par divers savants, Paris 33, No. 2, 1--128 (1908; JFM 39.0749.02); \textit{R. Bricard}, Mémoire sur les déplacements à trajectoires sphériques. Journal de l'École Polytechnique 11, No. 2, 1--96 (1906; JFM 37.0705.04)]. Some classes of Borel-Bricard motions are known, but their classification is far from complete. In this paper we have determined all self-motions of the `original' S-G platform. As a byproduct new types of Borel-Bricard motions have been discovered. During this task one has to use a computer for three different purposes. At first algebraic equations of self-motions have to be determined. For this purpose, we have used Maple V on a workstation. The use of a workstation is necessary, because we had to handle expressions having up to 20,000 terms. After equations of the self-motion have been derived one has to visualize the result, which means to plot trajectories of points of the platform and corresponding positions of the upper base. Here, we have two possibilities. If the motion can be parameterized, we can plot trajectories by a suitable graphical system. If the motion cannot be parameterized (this can happen because it is algebraic of order up to 8), we have to compute numerically some locations of the upper base of the platform and then to interpolate them. We did not go tot he last step because this would be beyond the scope of the paper. As a result we have shown that self-motions of the `original' S-G platform can be translatory motions, pure rotations, generalized screw motions with fixed axis, spherical four-bar mechanisms or more general space motions. One of the special cases of the general self-motion is a space motion with three circular trajectories, which is not spherical. This motion yields a new spatial mechanism, which is a spatial analogy of the well-known planar or spherical four-bar. It seems that Bricard already knew about the existence of such a motion. | 1 |
Octahedral mechanisms consisting of rigid rods in the edges of an octahedron connected by spherical joints at the vertices of the octahedron were subject to intense studies by Bricard, Blaschke, Wunderlich and others. Recently these mechanisms attracted again attention in the study of Stewart-Gough parallel platform mechanisms [see \textit{A. Karger} and \textit{M. Husty}, Comput.-Aided Des. 30, No. 3, 205--215 (1998; Zbl 0908.68187)]. Because of the number of constraints these mechanisms are in general rigid. Bricard, Blaschke and recently Husty and Karger [loc. cit.] were interested in special geometries that allow finite motion of the mechanism. By extending the motion group to equiform motions every such mechanism has at least one degree of freedom. In the paper the authors study the motion related to an octahedral mechanism in the group of equiform motions and give a local parametrization of the motion. Presents an integrated lesson on planning a garden and planting seeds with a focus on number concepts and Fibonacci numbers. (ERIC) | 0 |
The authors use the method, first applied by \textit{V. Berinde} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 18, 7347--7355 (2011; Zbl 1235.54024)], to obtain results about \(m\)-order fixed points and coincidence and common \(m\)-order fixed points of cyclic type for mappings depending on \(m\) variables in cone metric spaces. Let \(X\) be a complete metric space with metric \(d\), which is partially ordered. A mapping \(F: X\times X\to X\) is called mixed monotone if \(F(x,y)\) is monotone nondecreasing in \(x\) and monotone nonincreasing in \(y\). A pair \((\overline x,\overline y)\in X\times X\) is called a coupled fixed point of \(F\) if \(F(\overline x,\overline y)=\overline x\), \(F(\overline y,\overline x)=\overline y\). The main result of the paper is the following theorem.
Theorem. Let \(X\) be a partialy ordered complete metric space, let \(F: X\times X\to X\) be mixed monotone and such that
(i) There is a constant \(k\in [0,1)\) such that for each \(x\geq u\), \(y\leq v\)
\[
d(F(x,y), F(u,v))+ d(F(y, x), F(v,u))\leq k[d(x, u)+ d(y,v)].
\]
(ii) There exist \(x_0,y_0\in X\) with
\[
x_0\leq F(x_0, y_0)\quad\text{and}\quad y_0\leq F(y_0, x_0)
\]
or
\[
x_0\geq F(x_0, y_0)\quad\text{and}\quad y_0\leq F(y_0, x_0).
\]
Then \(F\) has a coupled fixed point \((\overline x,\overline y)\).
The author also gives conditions under which there exists a unique coupled fixed point. Finally, he applies this theorems to the periodic boundary value problem
\[
u'= h(t,u),\quad t\in (0,T),\quad u(0)= u(T)
\]
with \(h(t,u)= f(t,u)+ g(t,u)\). | 1 |
The authors use the method, first applied by \textit{V. Berinde} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 18, 7347--7355 (2011; Zbl 1235.54024)], to obtain results about \(m\)-order fixed points and coincidence and common \(m\)-order fixed points of cyclic type for mappings depending on \(m\) variables in cone metric spaces. While nearly all previous algorithms designed to solve simulation optimization problems have treated the outputs of simulation systems at a given design point (input parameter) as being independent of each other, this premise is flawed in that simulated outputs are generally correlated. We propose a decorrelation (DC) procedure that can effectively evaluate and remove the correlation of outputs of a simulation system. The proposed DC procedure is further integrated with STRONG, an improved framework of the well-known Response Surface Methodology (RSM), for tackling the simulation optimization problems with correlated outputs. This integration is particularly synergistic due to the fact that STRONG is a fully automated, response-surface-based procedure possessing appealing convergence properties and DC can take advantage of the concept of trust region as in STRONG to enable the removal of the correlation of outputs at the design points within the same trust region all at once. This is more efficient compared to the traditional approaches where a substantial number of observations are typically required for dealing with correlations. The resulting integrated method, which we call STRONG-DC, requires various adaptations so as to ensure the efficacy and efficiency of the overall framework. STRONG-DC preserves the desirable automation and convergence as STRONG, namely, it does not require human involvements and can be proved to achieve the truly optimal solution(s) with probability one (w.p.1) under reasonable conditions. Moreover, the effectiveness and efficiency of STRONG-DC are evaluated through extensive numerical analyses, along with a case study involving the well-known newsvendor problem. | 0 |
The book is the first volume of a two-volume exposition of the rapidly developing area of analysis related to integral operators in non-standard function spaces. The latter include variable exponent Lebesgue and amalgam spaces, variable Hölder spaces, variable exponent Campanato, Morrey and Herz spaces, Iwaniec-Sbordone (grand Lebesgue) spaces, grand variable exponent Lebesgue spaces, grand Morrey spaces, generalized grand Morrey spaces, and weighted analogues of some of them.
The main focus of the book is on mapping properties of diverse integral operators in these spaces.
The first volume includes the following chapters: 1. Hardy-type Operators in Variable Exponent Lebesgue Spaces. 2. Maximal, Singular, and Potential Operators in Variable Exponent Lebesgue Spaces with Oscillating Weights. 3. Kernel Integral Operators. 4. Two-weight Estimates. 5. One-sided Operators. 6. Two-weight Inequalities for Fractional Maximal Functions. 7. Description of the Range of Potentials, and Hypersingular Integrals. 8. More on Hypersingular Integrals and Embeddings into Hölder Spaces. 9. More on Compactness. 10. Applications to Singular Integral Equations.
The book is intended for researchers working in diverse branches of analysis and its applications. This is the second volume of the two-volume set devoted to the rapidly developing area of analysis related to integral operators in non-standard function spaces. The latter include variable exponent Lebesgue and amalgam spaces, variable Hölder spaces, variable exponent Campanato, Morrey and Herz spaces, Iwaniec-Sbordone (grand Lebesgue) spaces, grand variable exponent Lebesgue spaces, grand Morrey spaces, generalized grand Morrey spaces, and weighted analogues of some of them. As stated in Preface, this volume contains variable exponent results for Hölder, Morrey-Campanato, and grand spaces. In particular, it includes the problem of the boundedness of maximal, singular, and potential operators in variable exponent Morrey and Herz spaces and the case of unbounded underlying sets.
The volume consists of the following chapters (Chapters 1--10 are included in Volume 1):
11. Variable exponent Hölder spaces.
12. Morrey-type spaces; constant exponents.
13. Morrey, Campanato and Herz spaces with variable exponents.
14. Singular integrals and potentials in grand Lebesgue spaces.
15. Grand Lebesgue spaces on sets of infinite measure.
16. Fractional and singular integrals in grand Morrey spaces.
17. Multivariable operators on the cone of decreasing functions.
The book is intended for researchers working in diverse branches of analysis and its applications.
For Volume 1, see [the authors, Integral operators in non-standard function spaces. Volume 1: Variable exponent Lebesgue and Amalgam spaces. Basel: Birkhäuser/Springer (2016; Zbl 1385.47001)]. | 1 |
The book is the first volume of a two-volume exposition of the rapidly developing area of analysis related to integral operators in non-standard function spaces. The latter include variable exponent Lebesgue and amalgam spaces, variable Hölder spaces, variable exponent Campanato, Morrey and Herz spaces, Iwaniec-Sbordone (grand Lebesgue) spaces, grand variable exponent Lebesgue spaces, grand Morrey spaces, generalized grand Morrey spaces, and weighted analogues of some of them.
The main focus of the book is on mapping properties of diverse integral operators in these spaces.
The first volume includes the following chapters: 1. Hardy-type Operators in Variable Exponent Lebesgue Spaces. 2. Maximal, Singular, and Potential Operators in Variable Exponent Lebesgue Spaces with Oscillating Weights. 3. Kernel Integral Operators. 4. Two-weight Estimates. 5. One-sided Operators. 6. Two-weight Inequalities for Fractional Maximal Functions. 7. Description of the Range of Potentials, and Hypersingular Integrals. 8. More on Hypersingular Integrals and Embeddings into Hölder Spaces. 9. More on Compactness. 10. Applications to Singular Integral Equations.
The book is intended for researchers working in diverse branches of analysis and its applications. Ein Punkt gleicher Trägheit einer Figur ist ein Punkt von solcher Beschaffenheit, daß\ das Trägheitsmoment der Figur in bezug auf alle durch den Punkt gelegten Geraden konstant ist. Verf. hat früher (Sur les points d'égale inertie, les figures isotropes et les enveloppes d'égale inertie, Mémoires Acad. Bruxelles, 8\(^0\), 10 (1928), Nr. 2, 112 p. F. d. M. 57) bewiesen, daß\ das Dreieck zwei Punkte gleicher Trägheit besitzt, die auf der kleinen Achse der \textit{Steiner}schen Ellipse in einem Abstande \(d\) vom Mittelpunkt liegen, der durch die Gleichung \(d^2=\frac 18\gamma ^2\) gegeben ist, wo \(2\gamma \) den Abstand der Brennpunkte dieser Ellipse bedeutet. In der vorliegenden Arbeit berechnet Verf. die Koordinaten der Punkte gleicher Trägheit in bezug auf ein rechtwinkliges Koordinatensystem. Zuertst wird der Fall einer allgemeinen Lage dieses Koordinatensystems behandelt, dann werden zwei Sonderfälle untersucht, in denen der Ursprung des Koordinatensystems im Dreiecksschwerpunkt liegt. In dem einen Falle ist eine der Achsen einer Dreiecksseite, in dem anderen Falle einer Seitenhalbierenden parallel. Darauf wird die geometrische Konstruktion der Punkte gleicher Trägheit angegeben. Den Schluß\ bilden Untersuchungen über die Form der in verschiedenen Punkten der kleinen Achse der \textit{Steiner}ellipse konstruierten Trägheitsellipse. | 0 |
The article continues the author's joint paper with \textit{C. R. Hajarnavis} [Commun. Algebra 26, No. 6, 1985-1997 (1998; Zbl 0907.16006)]. It contains two interesting characterizations of fully bounded Noetherian Asano orders. A left or right ideal of a ring \(R\) is said to be integral if it contains a regular element of \(R\). If \(R\) admits a full quotient ring \(Q\), and every integral ideal of \(R\) has an inverse in \(Q\), then \(R\) is said to be an Asano order. Note that invertible ideals are left and right projective. If in addition, every one-sided integral ideal of \(R\) is projective, then \(R\) is said to be a Dedekind order. It is shown that every fully bounded Noetherian Asano order is a Dedekind order.
The main theorem asserts that an FBN ring \(R\) is a Dedekind order if and only if every integral maximal ideal \(M\) of \(R\) is localizable such that \(R_M\) is hereditary. Using this theorem, the author provides an internal characterization of Dedekind orders. Namely, an FBN ring \(R\) is a Dedekind order if and only if every integral maximal ideal of \(R\) is left and right projective and not idempotent. The theme of the paper is to extend the concepts of Asano order and Dedekind order from prime Noetherian rings to general Noetherian rings. In the prime case a non-zero ideal always contains a regular element, by Goldie's theorem. With this in mind it is natural, when considering a general Noetherian ring \(R\), to look at the integral ideals of \(R\) (i.e. those ideals which contain a regular element). After some preliminary results, the paper concentrates on the situation in which \(R\) is a fully bounded Noetherian ring with an over-ring \(T\) such that every integral ideal of \(R\) is invertible in \(T\). Initially it is not clear whether \(R\) has a classical quotient ring in which to take the inverses. However, it is convenient to ``shrink'' \(T\) if necessary so that it is the union of the inverses of the integral ideals of \(R\), and it is then shown that \(T\) is the quotient ring of \(R\). Also every one-sided ideal of \(R\) which contains a regular element is projective (which generalizes the fact that a fully bounded Noetherian prime Asano order is a Dedekind order), and if \(R\) has a non-zero Artinian one-sided ideals then \(R\) is a direct sum of prime rings. The finitely-generated torsion \(R\)-modules are studied, and it is shown that every factor ring of \(R\) by an integral ideal is Artinian with every one-sided ideal principal. | 1 |
The article continues the author's joint paper with \textit{C. R. Hajarnavis} [Commun. Algebra 26, No. 6, 1985-1997 (1998; Zbl 0907.16006)]. It contains two interesting characterizations of fully bounded Noetherian Asano orders. A left or right ideal of a ring \(R\) is said to be integral if it contains a regular element of \(R\). If \(R\) admits a full quotient ring \(Q\), and every integral ideal of \(R\) has an inverse in \(Q\), then \(R\) is said to be an Asano order. Note that invertible ideals are left and right projective. If in addition, every one-sided integral ideal of \(R\) is projective, then \(R\) is said to be a Dedekind order. It is shown that every fully bounded Noetherian Asano order is a Dedekind order.
The main theorem asserts that an FBN ring \(R\) is a Dedekind order if and only if every integral maximal ideal \(M\) of \(R\) is localizable such that \(R_M\) is hereditary. Using this theorem, the author provides an internal characterization of Dedekind orders. Namely, an FBN ring \(R\) is a Dedekind order if and only if every integral maximal ideal of \(R\) is left and right projective and not idempotent. Let \(X\) be a smooth projective variety of dimension \(n\). The Lefschetz standard conjecture in degree \(k\) predicts the existence of a codimension \(k\) rational cicle \(\mathcal{Z}_{\text{lef}}\in \operatorname{CH}^k(X\times X, \mathbb{Q})\) such that it induces an isomorphism \([\mathcal{Z}_{\text{lef}}]^*: H^{2n-k}(X, \mathbb{Q}) \rightarrow H^k(X, \mathbb{Q})\). This conjecture is important for different parts of the mathematics, for instance, it implies the variational Hodge conjecture and it allows realizing motivically the Lefschetz decomposition. This conjecture is known only in particular cases, for instance for abelian varieties and in degree two for many explicit families of hyper-Kähler manifolds.
In this paper, the author studies the Lefschetz standard conjecture in degree 2 for hyper-Kähler manifolds admitting a Lagrangian fibration or a Lagrangian covering. We remark that conjecturally every projective hyper-Kähler manifold is covered by lagrangian subvarieties. The author proves that a hyper-Kähler \(X\) equipped with a Lagrangian fibration and Picard number equals two as soon as it satisfies SYZ conjecture or it contains a divisor with negative square. She also proves this conjecture for a very general polarised hyper-Kähler fourfold covered by Lagrangian subvarieties. These proofs are based on a clever study of the maps induced in cohomology by cycles appearing in various situations. The last part of the paper is devoted to explaining the consequences of this conjecture for the structure of the Chow group and how the Bloch-Beilinson filtration precisely depends on the Lefschetz conjecture in degree 2. | 0 |
It is well known that the moduli space of polarized \(K3\) surfaces of a given degree \(2d\) is connected and the article under review investigates the case of irreducible holomorphic symplectic manifolds (IHSM) of \(K3^{[n]}\)-type. If \(X\) is such a manifold endowed with its natural quadratic form (the Bogomolov-Beauville form), let us fix \(h_d\in H^2(X,\mathbb{Z})\) a primitive element of degree \((h_d,h_d)=2d\) (for a given \(d>0\)) and let us consider the quantity (called divisibility)
\[
t:=\mathrm{div}(h_d)=[\mathbb{Z},(h_d,\, H^2(X,\mathbb{Z}))]
\]
i. e., the positive generator of the subgroup \((h_d,\, H^2(X,\mathbb{Z}))\). Standard considerations on discrimant of lattices show that \(t\mid\mathrm{gcd}(2d,2n-2)\). The main result of this paper consists in the computation of the number of connected components of the moduli space of IHSM of \(K3^{[n]}\)-type whose degree is \(2d\) and divisibility \(t\) (which has to satisfy the divisibility property above).
{ Theorem.} This number is equal to the number of isometry classes of pairs \((T,h)\) where \(T\) is an even positive definite lattice of rank 2 and discriminant \(\dfrac{4d(n-1)}{t^2}\), \(h\) a primitive element of square \(h^2=2d\) and such that \(h^\perp\) is generated by an element whose square is \(2n-2\).
The correspondence goes throught the Mukai lattice of a K3 (in which the Mukai lattice of \(X\) sits) and the study of the monodromy action (and more precisely of the parallel transport operators introduced by \textit{E. Markman} in [A survey of Torelli and monodromy results for holomorphic-symplectic varieties. Springer Proc. Math. 8, 257--322 (2011; Zbl 1229.14009)]. The article provides us with explicit computations of this number. An irreducible holomorphic symplectic (IHS) manifold is a simply connected compact Kähler manifold such that the space of holomorphic two forms is one-dimensional and spanned by an everywhere non-degenerate form. In dimension two these are precisely the \(K3\) surfaces which are usually studied via the Hodge structure on the second integral cohomology group.
One of the major results about \(K3\) surfaces is the Torelli theorem. Namely, two \(K3\) surfaces are isomorphic if and only if there exists a Hodge isometry between their second intergral cohomology groups. The paper under review surveys the recent results concerning a Torelli type theorem for IHS-manifolds.
The situation in higher dimensions is of course more difficult. For example, there exist birational IHS-manifolds which are not isomorphic and there exist non-birational IHS-manifolds whose second integral cohomology groups are Hodge-isometric.
Nevertheless, it is possible to formulate a Hodge theoretic Torelli theorem for IHS-manifolds, namely: Two deformation equivalent IHS-manifolds \(X, X'\) are bimeromorphic if and only if there exists a parallel-transport operator \(f: H^2(X,\mathbb{Z})\rightarrow H^2(X',\mathbb{Z})\) which is an isomorphism of integral Hodge structures. Here, an isomorphism \(\varphi: H^*(X,\mathbb{Z})\rightarrow H^*(X',\mathbb{Z})\) is a parallel-transport operator if, roughly speaking, the two manifolds in question are fibres in a family \(\pi : \mathcal{X}\rightarrow B\) over an analytic base and \(\varphi\) is induced by the parallel transport in the local system \(R\pi_*\mathbb{Z}\) along the path connecting the base points over which \(X\) and \(X'\) live. An isomorphism \(g: H^k(X,\mathbb{Z})\rightarrow H^k(X',\mathbb{Z})\) is a parallel transport operator if it is the \(k\)-th graded summand of a \(\varphi\) as above. If \(X=X'\), then a parallel transport operator is called a monodromy operator and it is the group of all monodromy operators, and some special subgroups of it, which is the focus of this paper. For example, one can consider \(\text{Mon}^2(X)\), the image of the group of monodromy operators in \(O(H^2(X,\mathbb{Z}))\), and its subgroup \(\text{Mon}_{\text{Hdg}}^2(X)\) of operators preserving the Hodge structure. One of the topics in the paper is the study of the subgroup \(W_{\text{Esc}}\), the isometry group of the weight 2 Hodge structure generated by reflection with respect to exceptional divisors, which turns out to be normal in \(\text{Mon}_{\text{Hdg}}^2(X)\). Other considered topics include, in particular, a proof of a weak version of Morrison's movable cone conjecture and a description of the moduli space of polarized holomorphic symplectic varieties as monodromy quotients of period domains of type IV. | 1 |
It is well known that the moduli space of polarized \(K3\) surfaces of a given degree \(2d\) is connected and the article under review investigates the case of irreducible holomorphic symplectic manifolds (IHSM) of \(K3^{[n]}\)-type. If \(X\) is such a manifold endowed with its natural quadratic form (the Bogomolov-Beauville form), let us fix \(h_d\in H^2(X,\mathbb{Z})\) a primitive element of degree \((h_d,h_d)=2d\) (for a given \(d>0\)) and let us consider the quantity (called divisibility)
\[
t:=\mathrm{div}(h_d)=[\mathbb{Z},(h_d,\, H^2(X,\mathbb{Z}))]
\]
i. e., the positive generator of the subgroup \((h_d,\, H^2(X,\mathbb{Z}))\). Standard considerations on discrimant of lattices show that \(t\mid\mathrm{gcd}(2d,2n-2)\). The main result of this paper consists in the computation of the number of connected components of the moduli space of IHSM of \(K3^{[n]}\)-type whose degree is \(2d\) and divisibility \(t\) (which has to satisfy the divisibility property above).
{ Theorem.} This number is equal to the number of isometry classes of pairs \((T,h)\) where \(T\) is an even positive definite lattice of rank 2 and discriminant \(\dfrac{4d(n-1)}{t^2}\), \(h\) a primitive element of square \(h^2=2d\) and such that \(h^\perp\) is generated by an element whose square is \(2n-2\).
The correspondence goes throught the Mukai lattice of a K3 (in which the Mukai lattice of \(X\) sits) and the study of the monodromy action (and more precisely of the parallel transport operators introduced by \textit{E. Markman} in [A survey of Torelli and monodromy results for holomorphic-symplectic varieties. Springer Proc. Math. 8, 257--322 (2011; Zbl 1229.14009)]. The article provides us with explicit computations of this number. We consider abstractly defined time series arrays \(y_t(T),\;1\leq t\leq T\), requiring only that their sample lagged second moments converge and that their end values \(y_{1+j}(T)\) and \(y_{T-j}(T)\) be of order less than \(T^{1/2}\) for each \(j\geq0\). We show that, under quite general assumptions, various types of arrays that arise naturally in time series analysis have these properties, including regression residuals from a time series regression, seasonal adjustments and infinite variance processes rescaled by their sample standard deviation.
We establish a useful uniform convergence result, namely that these properties are preserved in a uniform way when relatively compact sets of absolutely summable filters are applied to the arrays. This result serves as the foundation for the proof, in our companion paper, J. Econom. 118, 151--187 (2004; Zbl 1033.62092), of the consistency of parameter estimates specified to minimize the sample mean squared multistep-ahead forecast error when invertible short-memory models are fit to (short- or long-memory) time series or time series arrays. | 0 |
As in the paper by \textit{A. H. Ganchev} and \textit{W. Greenberg} [ibid. 17, No.1, 93-105 (1987; reviewed above)], let T be an injective, selfadjoint operator on the Hilbert space H, while A is now such that \(B=I-A\) is compact andpositive. The unique solvability of the abstract equation \((T\psi)'(x)=-A\psi (x),\) \(0<x<\infty\), with half-range boundary conditions at \(x=0\), is established under the only assumption that \(spr(B)<1\) (some misprints in the proof of Th. 2). It is then proved that, for spr(B)\(\leq 1\), the finite slab problem \((0<x<1\), say) is also uniquely solvable. These results, which are remarkable for their simplicity, are completed by some applications to linear transport (only sketches of the proofs are given). Let T, A be linear operators on the Hilbert space H. The linear abstract transport equation \((T\psi)'(x)=-A\psi (x),\) \(0<x<\infty\), with half- range boundary conditions at \(x=0\), is shown to have a unique solution \(\psi\) (x)\(\in H\) if the following assumptions are made: (i) T is injective and self-adjoint (ii) A is accretive, invertible and such that \(B=I-A\) is compact (iii) \(A^{-1}T\) has no eigenvalues on the imaginary axis (this holds, in particular, when \(Ker(Re A)=Ker A=0)\) (iv) \(\exists \alpha >0\) such that Ran \(B\subseteq Ran| T|^{\alpha}\cap D(| T|^{\alpha +1})\), where via the maximal positive and negative projectors for T. An application to the multigroup neutron transport equation with an unsymmetric scattering matrix is given. | 1 |
As in the paper by \textit{A. H. Ganchev} and \textit{W. Greenberg} [ibid. 17, No.1, 93-105 (1987; reviewed above)], let T be an injective, selfadjoint operator on the Hilbert space H, while A is now such that \(B=I-A\) is compact andpositive. The unique solvability of the abstract equation \((T\psi)'(x)=-A\psi (x),\) \(0<x<\infty\), with half-range boundary conditions at \(x=0\), is established under the only assumption that \(spr(B)<1\) (some misprints in the proof of Th. 2). It is then proved that, for spr(B)\(\leq 1\), the finite slab problem \((0<x<1\), say) is also uniquely solvable. These results, which are remarkable for their simplicity, are completed by some applications to linear transport (only sketches of the proofs are given). The authors start with the observation that the mean variance asset allocation rule does not incorporate the time horizon factor. First they prove the expected utility representing an investor's preferences can be approximated to any degree of accuracy by a finite sum of left and right tail probabilies of portfolios' returns. Next they study these tail probabilities by making use of certain asymptotic relationship with \(T\to \infty\) borrowed from the theory of large and moderate deviations. The authors succed in summarizing various time-varying properties of asset returns in terms of just one parameter introduced in the paper, the dilation exponent \(\alpha\). They show that country-specific factors overshadowed industry-specific ones, a fact known already from empirical findings. Finally, they demonstrate how \(\alpha\) may be used to enhance medium and long term investor's portfolio choices, both domestic and international. | 0 |
If \(N\) is a non-Abelian normal subgroup of a finite group \(G\) such that \(|G:C(x)|=k\) for all \(x\in N\setminus N\cap Z(G)\), then \(N^p\subseteq Z(G) \) for some prime \(p\). This is proved by the authors, generalizing a result of \textit{X. Zhao} and \textit{X. Guo} [Chin. Ann. Math., Ser. B 30, No. 4, 427-432 (2009; Zbl 1213.20031)] who assumed that \(N\) contains a non-central Sylow \(p\)-subgroup of \(G\) for some \(p\). The question put in the review mentioned is therefore answered. The authors use the classification of groups all of whose elements have prime power order to show solvability of \(N\). There have been many results over the last few years which relate the structure of groups to the sizes of conjugacy classes. The authors contribute to this study by proving the following interesting theorem: Let \(G\) be a finite group and \(R\) be a non-central normal subgroup of \(G\). Let \(N\) be a normal subgroup of \(G\) containing \(R\) such that each element of \(N\) has either \(1\) or \(m\) conjugates in \(G\), where \(m\) is some fixed natural number. Then \(N\) is nilpotent. By taking \(N\) to be the group \(G\), the old result of \textit{N. Itô} follows [Nagoya Math. J. 6, 17-28 (1953; Zbl 0053.01202)].
There are two comments which seem to the reviewer to be worth making. It is a straightforward result that in any transitive permutation group there is an element which acts without fixed points. In this paper there is no need to use the much stronger result that there is an element of prime-power order with this property, a result which depends on the classification of finite simple groups. The second comment is whether the condition that \(N\) contains a non-central Sylow subgroup is needed. The authors make no reference to this in the paper. | 1 |
If \(N\) is a non-Abelian normal subgroup of a finite group \(G\) such that \(|G:C(x)|=k\) for all \(x\in N\setminus N\cap Z(G)\), then \(N^p\subseteq Z(G) \) for some prime \(p\). This is proved by the authors, generalizing a result of \textit{X. Zhao} and \textit{X. Guo} [Chin. Ann. Math., Ser. B 30, No. 4, 427-432 (2009; Zbl 1213.20031)] who assumed that \(N\) contains a non-central Sylow \(p\)-subgroup of \(G\) for some \(p\). The question put in the review mentioned is therefore answered. The authors use the classification of groups all of whose elements have prime power order to show solvability of \(N\). By a perturbative argument, we construct solutions for a plasma-type problem with two opposite-signed sharp peaks at levels 1 and \(-\gamma\), respectively, where \(0<\gamma<1\). We establish some physically relevant qualitative properties for such solutions, including the connectedness of the level sets and the asymptotic location of the peaks as \(\gamma \rightarrow 0^+\). | 0 |
The authors establish the orthogonal stability of the Euler--Lagrange type functional equation
\[
f(mx + y) + f(mx - y) = 2f(x + y) + 2f(x - y) + 2(m^2 - 2)f(x) - 2f(y),
\]
where \(m\) is an arbitrary but fixed real constant with \(m \neq 0, \pm 1, \pm \sqrt{2}\) in orthogonality normed spaces controlled by the mixed type product-sum function \(\varphi(x, y)=\varepsilon(\| x\| ^p\| y\| ^p + \| x\| ^{2p}+\| y\| ^{2p})\).
The authors miss citing the recent papers on orthogonal stability, see \textit{M. Mirzavaziri} and \textit{M. S. Moslehian} [Bull. Braz. Math. Soc. (N.S.) 37, No. 3, 361--376 (2006; Zbl 1118.39015)] and references therein. The authors prove the orthogonal stability of the quadratic functional equation of Pexider type by using the fixed point alternative theorem. The following theorem is the main result of this paper: Suppose that \(X\) is a real orthogonality space with a symmetric orthogonal relation \(\perp\) and \(Y\) is a Banach space. Let the mappings \(f, g, h, k : X \rightarrow Y\) satisfy
\[
\|f (x + y) + g(x - y) - h(x) - k(y)\|\leq \varepsilon
\]
for all \(x, y \in X\) with \(x\perp y\). There exist an orthogonally additive mapping \(T : X\rightarrow Y\) and a constant \(C_1\geq 0\) such that
\[
\| f (x) - T (x)\| \leq C_1\varepsilon\quad (\text{for all }x \in X)
\]
if and only if there exists a constant \(C_2\geq 0\) with
\[
\|f (2x) - f (-2x) - 4f (x) - 4f (-x)\|\leq C_2\varepsilon\quad (\text{for all } x \in X).
\]
Indeed, if \(f : X\rightarrow Y\) satisfies
\[
\|f (2x) - f (-2x) - 4f (x) - 4f (-x)\|\leq \varepsilon\quad (\text{for all }x \in X),
\]
then there exist orthogonally additive mappings \(T_1 , T_2 , T_3 : X\rightarrow Y\) such that
\[
\begin{aligned}
\|f (x) - f (0) - T_1 (x)\| &\leq \frac {140}{3}\varepsilon,\\
\|g(x) - g(0) - T_2 (x)\|&\leq \frac {98}{3}\varepsilon,\\
\|h(x) + k(x) - h(0) - k(0) - T_3 (x)\|&\leq \frac {256}{3}\varepsilon \end{aligned}
\]
for all \(x \in X\).
In the Introduction, quoting the stability of quadratic equations of Pexider type, the authors missed citing a paper ``Stability of the quadratic equation of Pexider type'' [Abh. Math. Semin. Univ. Hamb. 70, 175--190 (2000; Zbl 0991.39018)] by \textit{S.-M. Jung}, which contains the first result about the stability of quadratic equations of Pexider type (see Theorem 5 and Corollary 6 in the paper). | 1 |
The authors establish the orthogonal stability of the Euler--Lagrange type functional equation
\[
f(mx + y) + f(mx - y) = 2f(x + y) + 2f(x - y) + 2(m^2 - 2)f(x) - 2f(y),
\]
where \(m\) is an arbitrary but fixed real constant with \(m \neq 0, \pm 1, \pm \sqrt{2}\) in orthogonality normed spaces controlled by the mixed type product-sum function \(\varphi(x, y)=\varepsilon(\| x\| ^p\| y\| ^p + \| x\| ^{2p}+\| y\| ^{2p})\).
The authors miss citing the recent papers on orthogonal stability, see \textit{M. Mirzavaziri} and \textit{M. S. Moslehian} [Bull. Braz. Math. Soc. (N.S.) 37, No. 3, 361--376 (2006; Zbl 1118.39015)] and references therein. We consider the following problem: Given ordered labeled trees \(S\) and \(T\), can \(S\) be obtained from \(T\) by deleting nodes? Deletion of the root node \(u\) of a subtree with children \(\langle T_1,\cdots,T_n\rangle\) means replacing the subtree by the trees \(T_1,\cdots,T_n\). The problem is motivated by the study of query languages for structured text data bases. The simple solutions to this problem require exponential time. We give an algorithm based on dynamic programming requiring \(O(|S||T|)\) time and space. | 0 |
Nonsymmetric Jack, Hermite and Laguerre polynomials occur as the polynomial part of the eigenfunction for certain many body Schrödinger operators with a pair potential proportional to \((1-s_{ij})/r^2_{ij}\), where \(s_{ij}\) denotes the coordinate exchange operator and \(r_{ij}\) denotes the separation between particles. The constant term normalization of the nonsymmetric Jack polynomials is evaluated using recurrence relations, and is related to the norm for the nonsymmetric analogue of the power-sum inner product. The results obtained for the nonsymmetric Hermite and Laguerre polynomials allow the explicit determination of the integral kernels which occur in Dunkl's theory of integral transforms based on reflection groups of type \(A\) and \(B\), and enable many analogues of properties of the classical Fourier, Laplace and Hankel transforms to be derived. The kernels are given as generalized hypergeometric functions based on nonsymmetric Jack polynomials. Central to the calculations is the construction of lowering-type operators for the nonsymmetric Jack polynomials of argument \(x\) and \(x^2\), which are the counterpart to the raising-type operator introduced by \textit{F. Knop} and \textit{S. Sahi} [Invent. Math. 128, No. 1, 9-22 (1997; Zbl 0870.05076)]. The paper deals with Jack polynomials \(J_\lambda(x;\alpha)\) and a similar family of (nonsymmetric) polynomials \(F_\lambda(x;\alpha)\), recently constructed by Opdam. Two characterizations are given: A recursion formula for \(F_\lambda\) and a combinatorial formula in terms of certain generalized tableaux for both \(F_\lambda\) and \(J_\lambda\). These characterizations are more explicit than the usual definitions as orthogonal family in the ring of symmetric functions, or as eigenfunctions of certain differential operators. Taking these characterizations as new definitions, existence of the polynomials is immediately clear; moreover, the conjecture of Macdonald (concerning the coefficients \(v_{\lambda\mu}(\alpha)\) of \(m_\mu\) in the expansion of \(J_\lambda(x;\alpha)\)) follows easily from the combinatorial formula. | 1 |
Nonsymmetric Jack, Hermite and Laguerre polynomials occur as the polynomial part of the eigenfunction for certain many body Schrödinger operators with a pair potential proportional to \((1-s_{ij})/r^2_{ij}\), where \(s_{ij}\) denotes the coordinate exchange operator and \(r_{ij}\) denotes the separation between particles. The constant term normalization of the nonsymmetric Jack polynomials is evaluated using recurrence relations, and is related to the norm for the nonsymmetric analogue of the power-sum inner product. The results obtained for the nonsymmetric Hermite and Laguerre polynomials allow the explicit determination of the integral kernels which occur in Dunkl's theory of integral transforms based on reflection groups of type \(A\) and \(B\), and enable many analogues of properties of the classical Fourier, Laplace and Hankel transforms to be derived. The kernels are given as generalized hypergeometric functions based on nonsymmetric Jack polynomials. Central to the calculations is the construction of lowering-type operators for the nonsymmetric Jack polynomials of argument \(x\) and \(x^2\), which are the counterpart to the raising-type operator introduced by \textit{F. Knop} and \textit{S. Sahi} [Invent. Math. 128, No. 1, 9-22 (1997; Zbl 0870.05076)]. Die theoretische Untersuchung des Hrn. Duhem (siehe JFM 20.1209.01) betrifft die von Hrn. Cailletet im Jahre 1880 beobachteten Erscheinungen bei der Verdichtung eines Gemenges von 1 Teil Luft und 5 Teilen Kohlensäure. Dieses letztere Gas wurde zunächst bei mässigem Drucke flüssig. Wenn dann, damit die Temperatur constant bliebe, der Druck langsam vergrössert wurde, so verschvand die Flüssigkeit für einen hinlänglichen Druck. Bei langsamer Verminderung des Druckes erscheint darauf die Flüssigkeit plötzlich wieder in dem Augenblick, wenn man bei dem Drucke anlangt, für den sie bei dem ersten Versuch verschwunden war, und bei einer gegebenen Temperatur bildet sich der Meniskus aus, sobald der Druck einen bestimmten Wert erreicht hat, der um so tiefer liegt, je höher die Temperatur ist. Die von Jamin (Almeida J. (2) VII. 389ff.) vorgeschlagene Theorie verwirft Hr. Duhem. Dagegen leitet er eine neue aus seinen Formeln ab, die er in der Schrift: ``Sur le potentiel thermodynamique'' aufgestellt hat. Zum Schlusse bemerkt er: ``Die Gleichung \(F(p, T, \lambda ) =0,\) auf deren Betrachtung diese ganze Theorie beruht, nimmt augenscheinlich eine andere Gestalt an, wenn man verschiedene Gase betrachtet. Ein Gemisch aus Wasserstoff und Kohlensäure wird sich also anders verhalten als ein Gemisch aus Luft und Kohlensäure''. In dem Berichte, den Hr. Duhem über die Versuche von Andrews erstattet (welche sich auf Gemenge von Kohlensäure und Stickstoff erstreckten, und welche, nach dem Tode von Andrews veröffentlicht, Hrn. Duhem bei der Abfassung seiner Arbeit noch nicht bekannt waren), wird die vollkommene Uebereinstimmung dieser Versuche mit der gegebenen Theorie festgestellt. | 0 |
The paper is a developing of the second author's paper [SIAM J. Numer. Anal. 40, 872-890 (2002; Zbl 1053.65062)] devoted to the investigation of projected and half-explicit Runge-Kutta methods being applied to the initial value problem for autonomous index 2 differential-algebraic equations (DAEs) in Hessenberg form. Here the analogous problem is considered for the nonautonomous system \(\dot {u} = f\left( {u,p,\lambda} \right),\quad 0 = g\left( {u,p} \right)\) where \(p = p\left( {t,p_{0}} \right)\) is the solution of the problem \(\dot {p} = k\left( {p} \right),\quad p\left( {0} \right) = p_{0} \). The solution \(p\left( {t,p_{0}} \right)\) belongs (on assumption) to a compact \(P\). It is shown (under some analytical and algebraic conditions) that the cocycle structure of the DAE is preserved under discretization.
If the DAE has a uniform (forward or pullback) attractor, then the corresponding discrete-time system also has a uniform attractor. Moreover, the component subsets of the numerical attractors converge upper semicontinuously to their counterparts uniformly in \(p\). The theoretical results are illustrated by two examples connected with constrained optimization and a ring modulator calculation. The paper is devoted to the investigation of projected and half-explicit Runge-Kutta (RK) methods being applied to the initial problem for index 2 differential-algebraic equations (DAEs) in Hessenberg form \(\dot {u} = f\left( {u,\lambda} \right),\quad 0 = g\left( {u} \right)\). The asymptotic features of the numerical and the exact solutions are compared. It is shown that the mentioned variants of the RK methods correctly reproduce the geometric properties of the continuous system. Namely, the author presents analytical and algebraic conditions (for each variant other) providing (under sufficiently small step) existence of the infinite sequence of the numerical solution and its stability with respect to the solutions manifold. Besides, if the DAE has a hyperbolic periodic solution then the \(u\)-component of the corresponding discrete dynamics is passing in a small vicinity of the orbit's \(u\)-component. | 1 |
The paper is a developing of the second author's paper [SIAM J. Numer. Anal. 40, 872-890 (2002; Zbl 1053.65062)] devoted to the investigation of projected and half-explicit Runge-Kutta methods being applied to the initial value problem for autonomous index 2 differential-algebraic equations (DAEs) in Hessenberg form. Here the analogous problem is considered for the nonautonomous system \(\dot {u} = f\left( {u,p,\lambda} \right),\quad 0 = g\left( {u,p} \right)\) where \(p = p\left( {t,p_{0}} \right)\) is the solution of the problem \(\dot {p} = k\left( {p} \right),\quad p\left( {0} \right) = p_{0} \). The solution \(p\left( {t,p_{0}} \right)\) belongs (on assumption) to a compact \(P\). It is shown (under some analytical and algebraic conditions) that the cocycle structure of the DAE is preserved under discretization.
If the DAE has a uniform (forward or pullback) attractor, then the corresponding discrete-time system also has a uniform attractor. Moreover, the component subsets of the numerical attractors converge upper semicontinuously to their counterparts uniformly in \(p\). The theoretical results are illustrated by two examples connected with constrained optimization and a ring modulator calculation. Asynchronous circuits can efficiently interconnect system-on-chip modules with different clock domains. Fulcrum's Nexus Interconnect features a 16-port, 36-bit asynchronous crossbar that connects through asynchronous channels to clock-domain converters for each synchronous module. In TSMC'S 130-nm process, Nexus achieves 1.35 GHz and transfers 780 Gbps. | 0 |
The author investigates reversible second order Hamiltonian systems of the form (HS) \(\ddot q + W_q(q,t)=f(t)\). Basic assumptions in addition to reversibility are that \(W \in C^2 (\mathbb{R}^n \times \mathbb{R}, \mathbb{R})\) is \(\mathbb{Z}^n\)-periodic in \(q \in \mathbb{R}^n\) and that \(W\) and \(f \in C (\mathbb{R}, \mathbb{R}^n)\) are periodic in \(t\) (the periodicity assumption can be weakened in order to cover different behavior for \(t > 0\) or \(t < 0)\). Two types of results are proved. The first type yields the existence of heteroclinic solutions between certain periodic orbits of minimal energy. The second yields a solution through a given \(\xi \in \mathbb{R}^n\) whose \(\omega\)-limit set is a periodic orbit of minimal energy. The case \(n = 1\) is studied more closely. The proofs are based on the calculus of variations. The paper is a sequel to an earlier paper of the author [Ergodic Theory Dyn. Syst. 14, No. 4, 817-829 (1994; Zbl 0818.34025)]. The author studies the Hamiltonian system (HS) \(\ddot q+ W_ q(t, q)= f(t)\), where \(W\in {\mathcal C}^ 2(\mathbb{R}\times \mathbb{R}^ n, \mathbb{R})\) and \(f\in {\mathcal C}(\mathbb{R}, \mathbb{R}^ n)\) are 1-periodic in all variables, \(f\) has mean value 0 and
\[
W(- t, q)+ f(- t, q)= W(t, q)+ f(t, q).
\]
Let \({\mathcal K}_ 1\) be the set of 1-periodic solutions of (HS) of minimal energy. The main result is the following. If \({\mathcal K}_ 1\) consists of isolated points (so (HS) cannot be autonomous) then for each \(\overline q\in {\mathcal K}_ 1\) there exists a heteroclinic solution \(Q\) connecting \(\overline q\) to some \(\widehat q\in {\mathcal K}_ 1- \{\overline q\}\), i.e. \(Q(t)- \overline q(t)\to 0\) as \(t\to -\infty\) and \(Q(t)- \widehat q(t)\to 0\) as \(t\to\infty\). The heteroclinic orbit is obtained by minimizing a certain functional. | 1 |
The author investigates reversible second order Hamiltonian systems of the form (HS) \(\ddot q + W_q(q,t)=f(t)\). Basic assumptions in addition to reversibility are that \(W \in C^2 (\mathbb{R}^n \times \mathbb{R}, \mathbb{R})\) is \(\mathbb{Z}^n\)-periodic in \(q \in \mathbb{R}^n\) and that \(W\) and \(f \in C (\mathbb{R}, \mathbb{R}^n)\) are periodic in \(t\) (the periodicity assumption can be weakened in order to cover different behavior for \(t > 0\) or \(t < 0)\). Two types of results are proved. The first type yields the existence of heteroclinic solutions between certain periodic orbits of minimal energy. The second yields a solution through a given \(\xi \in \mathbb{R}^n\) whose \(\omega\)-limit set is a periodic orbit of minimal energy. The case \(n = 1\) is studied more closely. The proofs are based on the calculus of variations. The paper is a sequel to an earlier paper of the author [Ergodic Theory Dyn. Syst. 14, No. 4, 817-829 (1994; Zbl 0818.34025)]. For the interaction of elderly people with IT systems, an ergonomic and intuitive design as well as self-explanatory handling processes are particularly relevant. Herein adequate acoustic feedback, which accounts for the specific needs and experience of the target group, provides high efficacy and acceptance of technology with regard to Human-Computer Interaction. In this study, five different types of sound schemes are evaluated on their intuitive understanding and memorization by older users. The participants assign audible feedback to typical applications of telemedical monitoring and have to reminisce given classifications. This approach makes it possible to elicit the homogeneity of psychoacoustic models of elderly people and give recommendations for the design of acoustic feedback mechanisms for this audience. As a result, the use of familiar sounds from everyday situations has been found significantly better in terms of the consistency of the intuitive mapping and memorization for use cases in a telemedical context, in comparison to synthetic sounds that obtain their semantic denotation just by convention. | 0 |
Let \(r(n)\) be the product of the distinct prime factors of \(n\). The author conjectures that if \(a+b= c\), with coprime positive integers \(a< b< c\), then \(r(abc)\geq c^{2/3}\) except for 11 specific numerical examples. This is an effective form of a weak \(abc\)-conjecture. From this he deduces that if \(A< B\) have the same prime factors, then \(B-A> A^{2/5}\), except when \(A= 3^{11}.23.109^2\) and \(B= 3.23^6.109\) or \(A= 2^2. 3^8.5.7\) and \(B= 2.3.5^5.7^7\). This provides a conjectural answer to a question of \textit{T. Cochrane} and \textit{R. E. Dressler} [Math. Comput. 68, 395-401 (1999; Zbl 0929.11031)]. Dressler conjectured that there is a prime between any two positive integers \(a<c\) with the same prime factors. The authors show herein that if the \(abc\)-conjecture is true then \(c-a> a^{1/2-o(1)}\). Moreover it is believed that the maximal gap between consecutive primes \(\leq x\) is \(O(\log^2x)\). If both conjectures hold then Dressler's conjecture holds for sufficiently large \(a<c\). The authors note that there are many examples of \(c-a<a^{1/2}\) but conjecture that \(c-a>a^{1/3}\) always. They prove \(c-a\gg_\varepsilon (\log a)^{3/4-\varepsilon}\). They also carefully consider the case where \(a\) and \(c\) have only two prime factors. | 1 |
Let \(r(n)\) be the product of the distinct prime factors of \(n\). The author conjectures that if \(a+b= c\), with coprime positive integers \(a< b< c\), then \(r(abc)\geq c^{2/3}\) except for 11 specific numerical examples. This is an effective form of a weak \(abc\)-conjecture. From this he deduces that if \(A< B\) have the same prime factors, then \(B-A> A^{2/5}\), except when \(A= 3^{11}.23.109^2\) and \(B= 3.23^6.109\) or \(A= 2^2. 3^8.5.7\) and \(B= 2.3.5^5.7^7\). This provides a conjectural answer to a question of \textit{T. Cochrane} and \textit{R. E. Dressler} [Math. Comput. 68, 395-401 (1999; Zbl 0929.11031)]. This paper proposes a practical method of determining the spanning matrix of any decomposable element in \({V^{(3)}}\), the space of 3rd-order completely symmetrical tensors, under each of four different cases. This method is routinized and formulated. | 0 |
Let \(R\) be a commutative ring and let \(T:(R\text{-modules})^{op}\times (R\text{-modules})\to (R\text{-modules})\) of bidegree \((k_1,k_2)\). For example, the bifunctor
\[
S(R^n,R^m)= H_0(\sum_{k_1}\times \sum_{k_2};\Hom((R^m)^{\otimes k_1},(R^n)^{\otimes k_2}\otimes A))
\]
has bidegree \((k_1,k_2)\).
The author describes a method for computing the stable K-theory of \(R\) with coefficients in \(T\), or rather the topological Hochschild homology, \(THH_*(R,T)\) [the author and \textit{T. Pirashvili}, J. Pure Appl. Algebra 96, No. 3, 245-258 (1994; Zbl 0812.19002)]. The method is to reduce to calculations where \(T\) is replaced by bifunctors like \(S\) and by bifunctors of lower bidegree. The method is illustrated in the case of \(R=\mathbb{F}_p\) and \(R=\mathbb{Z}\). The authors consider stable \(K\)-theory of a ring \(R\) as a functor \(K^ s_ *(R,-)\): \({\mathcal B} \to {\mathcal A}b\), where \({\mathcal B}\) is the abelian category of functors \(T:{\mathcal R}^{op}_ f \times {\mathcal R}_ f \to {\mathcal A}b\), and \({\mathcal R}_ f\) is the additive category of finitely generated, free \(R\)-modules.
The category \({\mathcal B}\) has a set of projective generators, \(\{P^{n,m} | n, m \in \mathbb{N} \cup \{0\}\}\), so the situation is amenable for classical homological algebra.
The authors' Conjecture A states that the groups \(K^ s_ i(R,P^{n,m})\) vanish for all \(n,m\) and all \(i > 0\). Using basic stability results of \textit{W. van der Kallen} [Homology stability for linear groups, Invent. Math. 60, 269-295 (1980; Zbl 0415.18012)], the authors verify the conjecture for any semi-simple ring and any commutative integral domain of finite Krull dimension. Furthermore, Conjecture A is shown to imply that \(K^ s_ i(R,-)\) is the \(i\)-th derived functor of \(K^ s_ 0(R,-)\).
Motivated by the description of \(THH_ * (R,M)\) as the homology groups of the small category \({\mathcal R}_ f\) with coefficients in \(\Hom (-,M \otimes_ R-) \in {\mathcal B}\), the authors define topological Hochschild homology \(THH_ *(R,T) : = H_ *({\mathcal R}_ f,T)\), for any \(T \in {\mathcal B}\). There results a natural transformation \(\nu : K^ s_ * (R,- ) \to THH_ *(R,-)\). The rest of the paper deals with the question: In which cases is \(\nu\) an isomorphism?
Basically, I find the approach of this paper quite attractive. However, at least in a couple of spots, more precision in statements and care in proofs would have made for easier reading.
First of all, I object to the use (in theorem 2) of ``='' to indicate the existence of a (not even natural) isomorphism between the abelian groups in question. A better statement of theorem 2 appears to be that ``the spectral sequence in question collapses and involves no extension problem.''
In the proof of theorem 2, the authors do mention a fair amount of relevant facts, but the reading is hard, not only because of the sloppy formulation of the theorem, but also because the authors fail to mention that the spectral sequence behaves well viz-a-viz the long exact sequences induced by a short exact sequence of coefficient functors.
In fact, if the authors had stated the theorem precisely and isolated the salient ``naturality'' of the spectral sequence clearly, some of the details actually given could probably have been safely left to the reader so that the overall length of the exposition would not have increased, while the reading would have been much easier.
Next, I have a question about the orbit set, \([V]\), introduced at the beginning of section 1. Presumably, in many examples, the apparent dependence on the dimension \(i\) (which is never mentioned) is real. Therefore, a comment concerning the behaviour under stabilization would have been nice to have. As it is, the authors have chosen a wording which -- if taken literally -- implies that \([V]\) is independent of \(i\).
Finally, I spotted a few misprints that are slightly disturbing: In theorem 1, ``\({\mathcal B}\):'' is missing. On page 250, statement (i) deals with \(E^ \infty_{p,q}\) rather than \(E^ 2_ \infty\). In lemma 1, an extra condition, \(K^ s_ i (R,P) = 0\) for \(i>0\), was intended. | 1 |
Let \(R\) be a commutative ring and let \(T:(R\text{-modules})^{op}\times (R\text{-modules})\to (R\text{-modules})\) of bidegree \((k_1,k_2)\). For example, the bifunctor
\[
S(R^n,R^m)= H_0(\sum_{k_1}\times \sum_{k_2};\Hom((R^m)^{\otimes k_1},(R^n)^{\otimes k_2}\otimes A))
\]
has bidegree \((k_1,k_2)\).
The author describes a method for computing the stable K-theory of \(R\) with coefficients in \(T\), or rather the topological Hochschild homology, \(THH_*(R,T)\) [the author and \textit{T. Pirashvili}, J. Pure Appl. Algebra 96, No. 3, 245-258 (1994; Zbl 0812.19002)]. The method is to reduce to calculations where \(T\) is replaced by bifunctors like \(S\) and by bifunctors of lower bidegree. The method is illustrated in the case of \(R=\mathbb{F}_p\) and \(R=\mathbb{Z}\). A key focus of current science education reforms involves developing inquiry-based learning materials. However, without an understanding of how working scientists actually do science, such learning materials cannot be properly developed. Until now, research on scientific reasoning has focused on cognitive studies of individual scientific fields. However, the question remains as to whether scientists in different fields fundamentally rely on different methodologies. Although many philosophers and historians of science do indeed assert that there is no single monolithic scientific method, this has never been tested empirically. We therefore approach this problem by analyzing patterns of language used by scientists in their published work. Our results demonstrate systematic variation in language use between types of science that are thought to differ in their characteristic methodologies. The features of language use that were found correspond closely to a proposed distinction between Experimental Sciences (e.g., chemistry) and Historical Sciences (e.g., paleontology); thus, different underlying rhetorical and conceptual mechanisms likely operate for scientific reasoning and communication in different contexts. | 0 |
Paralleling the process of defining almost paracomplex structures and paracomplex structures on smooth manifolds, which is a real version of almost complex structures and complex structures, the authors introduce the notions of almost para-CR structures and para-CR structures as a real version of almost CR-structures and CR-structures. To any point in a para-CR manifold, one can associate a fundamental graded Lie algebra. In their previous paper [Ann. Global Anal. Geom. 30, No.~1, 1--27 (2006; Zbl 1109.53030)], using the method of graded Lie algebras, the authors classified maximally homogeneous para-CR structures of semisimple type such that the associated graded semisimple Lie algebra has depth 2. In this paper, the authors classify all maximally homogeneous para-CR structures of semisimple type in terms of graded real semisimple Lie algebras. The authors define the notion of a (weak) almost para-CR structure on a manifold \(M\) as a pair \((HM,K)\) where \(HM\) is a rank],\(2n\) distribution on \(M\) and \(K\) is a field of endomorphisms of \(HM\) such that \(K^2 = id\) and \(K\neq\pm id\). If \(K\) satisfies some integrability conditions, then \((HM,K)\) is called a (weak) para-CR structure. It is shown that, under some regularity conditions, an almost para-CR structure can be identified with a Tanaka structure. Also, a classification of maximally homogeneous para-CR structures of semisimple type in terms of corresponding gradiations of real semisimple Lie algebras is given. Finally, there all such maximally homogeneous structures of depth two are listed and the integrability conditions are verified. | 1 |
Paralleling the process of defining almost paracomplex structures and paracomplex structures on smooth manifolds, which is a real version of almost complex structures and complex structures, the authors introduce the notions of almost para-CR structures and para-CR structures as a real version of almost CR-structures and CR-structures. To any point in a para-CR manifold, one can associate a fundamental graded Lie algebra. In their previous paper [Ann. Global Anal. Geom. 30, No.~1, 1--27 (2006; Zbl 1109.53030)], using the method of graded Lie algebras, the authors classified maximally homogeneous para-CR structures of semisimple type such that the associated graded semisimple Lie algebra has depth 2. In this paper, the authors classify all maximally homogeneous para-CR structures of semisimple type in terms of graded real semisimple Lie algebras. We develop a method for bias correction, which models the error of the target estimator as a function of the corresponding estimator obtained from bootstrap samples, and the original estimators and bootstrap estimators of the parameters governing the model fitted to the sample data. This is achieved by considering a number of plausible parameter values, generating a pseudo original sample for each parameter and bootstrap samples for each such sample, and then searching for an appropriate functional relationship. Under certain conditions, the procedure also permits estimation of the mean square error of the bias corrected estimator. The method is applied for estimating the prediction mean square error in small area estimation of proportions under a generalized mixed model. Empirical comparisons with jackknife and bootstrap methods are presented. | 0 |
[For part I, cf. J. Algebra 149, 179-182 (1992; Zbl 0761.11042).]
Let \(F\) be a finite field and \(S_ F\) the \(F\)-algebra of all semi- infinite sequences over \(F\). For a nonconstant polynomial \(f\) over \(F\) let \(S_ F (f(x))\) denote the set of all homogeneous linear recurring sequences in \(F\) with characteristic polynomial \(f\). For \({\mathbf s}, {\mathbf t}\in S_ F\) denote the convolution \(u_ n= \sum_{i=0}^ n s_ i t_{n-i}\) by \({\mathbf u}= {\mathbf s}*{\mathbf t}\). Denote by \(S_ F (f(x))* S_ F(g(x))\) the subspace of \(S_ F\) spanned by all such convolutions with \({\mathbf s}\in S_ F (f(x))\) and \({\mathbf t}\in S_ F(g(x))\). It is shown here that \(S_ F (xf(x))* S_ F(g(x))= S_ F(f(x) g(x))\) and also, by counterexample, that \(S_ F (f(x))* S_ F (g(x))\neq S_ F (f(x) g(x))\). Let \(F\) be a finite field and \(S_ F\) the \(F\) algebra of all sequences \({\mathbf s}=(s_ 0,s_ 1,\dots)\) over \(F\). For a nonconstant monic polynomial \(f(x)\) over \(F\), let \(S_ F(f(x))\) denote the set of all homogeneous linear recurring sequences over \(F\) with characteristic polynomial \(f(x)\). For \({\mathbf s},{\mathbf t}\in S_ F(f(x))\), define the convolution by \(u_ n=\sum_ i s_ i t_{n-i}\). Denote by \(S_ F(f(x))* S_ F(g(x))\) the subspace of \(S_ F\) spanned by all convolutions \({\mathbf s}*{\mathbf t}\), \({\mathbf s}\in S_ F(f(x))\), \({\mathbf t}\in S_ F(g(x))\). It is shown here that
\[
S_ F(f(x))* S_ F(g(x))\subseteq S_ F(f(x)g(x)).
\]
| 1 |
[For part I, cf. J. Algebra 149, 179-182 (1992; Zbl 0761.11042).]
Let \(F\) be a finite field and \(S_ F\) the \(F\)-algebra of all semi- infinite sequences over \(F\). For a nonconstant polynomial \(f\) over \(F\) let \(S_ F (f(x))\) denote the set of all homogeneous linear recurring sequences in \(F\) with characteristic polynomial \(f\). For \({\mathbf s}, {\mathbf t}\in S_ F\) denote the convolution \(u_ n= \sum_{i=0}^ n s_ i t_{n-i}\) by \({\mathbf u}= {\mathbf s}*{\mathbf t}\). Denote by \(S_ F (f(x))* S_ F(g(x))\) the subspace of \(S_ F\) spanned by all such convolutions with \({\mathbf s}\in S_ F (f(x))\) and \({\mathbf t}\in S_ F(g(x))\). It is shown here that \(S_ F (xf(x))* S_ F(g(x))= S_ F(f(x) g(x))\) and also, by counterexample, that \(S_ F (f(x))* S_ F (g(x))\neq S_ F (f(x) g(x))\). In recent decades, mathematics education scholarship has promoted a shift from traditional teaching and learning approaches to novel approaches that privilege peer-to-peer collaboration. To mobilize researchers towards advancing scholarship on collaborative learning, scholars need to understand the methodologies available to them for examining group work. In this study, I synthesize and critically examine discourse analytic methodologies for analyzing group work in mathematics. Using a systematic review methodology, an initial 576 articles were considered, and 31 articles met the inclusion criteria for analysis. Articles were separated according to their research foci and methodologies were synthesized and critically evaluated. The review provides researchers with methodological options for analyzing group discourse in mathematics. | 0 |
Given \(n\) blue points on the unit circle, this papers shows that is always possible to place at most a constant times \(r2^{\log^*r}\) red points, such that any convex set containing more than \(n/r\) blue points on the circle also contains one the red points inside. \((\log^*\) denotes the iterated logarithm function.)
Similar results are known in the case of any \(3d\) blue points and halfspaces with a bound of the number of red points needed of \(cr\), and in higher dimensions with halfspaces \(cdr \log (dr)\) [\textit{P. Haussler} and \textit{E. Welzl}, Discrete Comput. Geom. 2, 127-151 (1987; Zbl 0619.68056)]. \(c\) is a fixed constant. Moreover, then, the red points can be chosen as a subset of the blue points. The main problem may be described as follows: given a set of n points in d-dimensional Euclidean space, find a data structure that uses linear storage such that the number of points in any query half space can be determined in sublinear time \(O(n^{\alpha})\). A data structure with \(\alpha =d(d-1)/(d(d-1)+1)+\gamma\) for any \(\gamma >0\) is exhibited. These bounds for \(\alpha\) are better than those previously published for all \(d\geq 2\) by \textit{A. Yao} and \textit{F. Yao} [A general approach to d- dimensional geometric queries. Proc. 17th Symp. Theory of Computing, 163- 169 (1985)].
Let X be a set and R be a set of subsets of X, which have a finite dimension in Vapnik-Chervonenkis sense [\textit{V. N. Vapnik} and \textit{A. Ya. Chervonenkis}: The theory of pattern recognition (Russian) (1974; Zbl 0284.68070)], A be a finite subset of X and \(0\leq \epsilon \leq 1\). A subset N of A is an \(\epsilon\)-net of A (for R) if N contains a point in each \(r\in R\) such that \(| A\cap r| /| A| >\epsilon\). The authors prove that for \(0<\epsilon\), \(\delta <1\), if N is a subset of A obtained by \(m\geq \max (4/\epsilon \log 2/\delta,8d/\epsilon \log 8d/\epsilon)\) random independent draws, then N is an \(\epsilon\)-net of A with probability at least 1-\(\delta\). Using this result, a partition tree structure that achieves the above query time is constructed. | 1 |
Given \(n\) blue points on the unit circle, this papers shows that is always possible to place at most a constant times \(r2^{\log^*r}\) red points, such that any convex set containing more than \(n/r\) blue points on the circle also contains one the red points inside. \((\log^*\) denotes the iterated logarithm function.)
Similar results are known in the case of any \(3d\) blue points and halfspaces with a bound of the number of red points needed of \(cr\), and in higher dimensions with halfspaces \(cdr \log (dr)\) [\textit{P. Haussler} and \textit{E. Welzl}, Discrete Comput. Geom. 2, 127-151 (1987; Zbl 0619.68056)]. \(c\) is a fixed constant. Moreover, then, the red points can be chosen as a subset of the blue points. In this work, numerical simulations are performed and spatial distributions of specific parameters of nonlinear focused ultrasound beams of various geometry are compared. The numerical algorithm is based on the solution of the Khokhlov-Zabolotskaya (KZ) equation. Focused acoustic beams of periodic waves with an initially uniform amplitude distribution, typical for medical therapeutic transducers, and with Gaussian amplitude shading are considered. Numerical solutions are obtained and analyzed for nonlinear acoustic field in various regimes of linear, quasilinear, and nonlinear propagation when shock fronts are developed in the waveform close to the focus and while propagating to the focus of the beam.{\par\copyright 2008 American Institute of Physics} | 0 |
[For part I see the author in ibid. 61, 33-65 (1987; Zbl 0642.33017).]
The author continues his attempts to find generalizations of Tuck's integral
\[
\int_{-1}^ 1 {{P_ n(x)- P_ n(t)} \over {| x- t|}} dt= 2H_ n P_ n(x),
\]
\(P_ n(x)\) the Legendre polynomial and \(H_ n= 1+{1\over 2}+ \cdots+ {1\over n}\). Here he looks for a kernel of the form \(\Phi(x,t)= g(x^ 2-2 \mu xt+ t^ 2)\), and uses ultraspherical polynomials. The function he finds is \(g(t)= | t+\mu^ 2 - 1|^{-\nu}\), \(\nu>0\), \(g(t)= {1\over 2}\log| t+\mu^ 2-1|\) for \(\nu=0\). The expansion of the kernel has two parts. One is the well known one which follows from the integrated form of the addition formula. The second is related to a weighted Hilbert transform of the first expansion. The author gives a proof that
\[
(A)\quad \int^{1}_{-1}((C^{\nu}_ n(x)-C^{\nu}_ n(t))/| x-t|^{2\nu})(1-t^ 2)^{\nu -} dt = \pi (\cos \nu \pi)^{-1}[1-(2\nu)_ n/n!]C^{\nu}_ n(x),
\]
-1\(\leq x\leq 1\), \(n=1,2,...\), \(-<\nu <1\). For \(\nu = 1/2\) this was found by E. O. Tuck. For \(-1/2 < \nu < 0\) this identity is shown to follow from an expansion found by Pólya and Szegö. A different, direct proof is given for \(-<\nu <1\). For \(\nu =0\) the direct limiting case is
\[
\int^{1}_{-1}T_ n(x)-T_ n(t)(1-t^ 2)^{-1/2} dt = \pi T_ n(x),\quad -1\leq x\leq 1,
\]
which is different than an earlier result of the author. The author's result can be obtained from (A) and
\[
(B)\quad \int^{1}_{-1}C^{\nu}_ n(x)(1-t^ 2)^{\nu-1/2} dt = C^{\nu}_ n(x)\Gamma (\nu +1/2)\Gamma (1/2)/\Gamma (\nu +1)
\]
by subtracing (B) from (A), dividing by \(\nu^ 2\) and letting \(\nu\to 0\). | 1 |
[For part I see the author in ibid. 61, 33-65 (1987; Zbl 0642.33017).]
The author continues his attempts to find generalizations of Tuck's integral
\[
\int_{-1}^ 1 {{P_ n(x)- P_ n(t)} \over {| x- t|}} dt= 2H_ n P_ n(x),
\]
\(P_ n(x)\) the Legendre polynomial and \(H_ n= 1+{1\over 2}+ \cdots+ {1\over n}\). Here he looks for a kernel of the form \(\Phi(x,t)= g(x^ 2-2 \mu xt+ t^ 2)\), and uses ultraspherical polynomials. The function he finds is \(g(t)= | t+\mu^ 2 - 1|^{-\nu}\), \(\nu>0\), \(g(t)= {1\over 2}\log| t+\mu^ 2-1|\) for \(\nu=0\). The expansion of the kernel has two parts. One is the well known one which follows from the integrated form of the addition formula. The second is related to a weighted Hilbert transform of the first expansion. We study the quantization of chiral QED with one family of massless fermions and the Stueckelberg field in order to give mass to the Abelian gauge field in a BRST-invariant way. We show that an extended Slavnov-Taylor (ST) identity can be introduced and fulfilled to all orders in perturbation theory by a suitable choice of the local actionlike counterterms, order by order in the loopwise expansion. We discuss the physical content of the extended ST identity and prove that the cohomology classes associated with \({\mathcal S}_0'\) are modified with respect to the ones of the classical BRST differential \(s\) in the FP neutral sector (physical observables). We explicitly check that the physical states defined by \(s\) are no more physical states of the full quantized theory by showing that the subspace of the physical states corresponding to \(s\) is not left-invariant under the application of the \(S\) matrix, as a consequence of the extended ST identity. | 0 |
As well known the classical uni-dimensional Fourier transform is a very powerful tool which is very useful in Analysis as well as in many other areas of mathematics. It is also used in many practical applications like information storage procedures, for instance.
There have been several generalizations of the classical Fourier transform to higher dimensions like the hypercomplex Fourier or the Quaternionic Fourier transform which were introduced by Bülow and Sommer. One type of these generalizations is formed by the Clifford Fourier transforms, which are based on Clifford algebra on \(\mathbb R^m\) and Clifford analysis, whose basic notions are briefly and neatly introduced in the paper under review. One of the main difficulties of the study of these generalizations is the non-commutativity of Clifford Algebra and finding a kernel that allow one to work with the transform to extend the results of the uni-dimensional classical Fourier Transform.
The authors of the present paper in [``The Clifford-Fourier transform'', J. Fourier Anal. Appl. 11, No. 6, 669--681 (2005; Zbl 1122.42015)] introduced one of these types of Clifford transforms. This transform can be defined by
\[
(\mathcal F f)(y)={1\over 2\pi} \int_{\mathbb R^2} e^{x \wedge y} f(x)\,dA(x),
\]
where \(f\) is an integrable complex function on \(\mathbb R^2\). Originally, it was defined by decomposing the uni-dimensional Fourier transform a an exponential of a scalar operator involving the Laplacian. All these facts are clearly presented in the paper under review. Indeed, the concepts are motivated and kindly introduced so that even someone who is not familiar with the subject can read the paper. Furthermore, a deep study of their Clifford Fourier transform is done. In particular, thanks to the knowledge of the kernel of the transform, they are able to show that the classical results, like the inversion and the convolution theorem, extend to their Clifford Fourier transform. Also, they study other properties like relating their transform to the standard tensorial Fourier transform. The authors thus propose their generalization for higher dimensions. They compute Clifford Fourier transform of the characteristic function and give some application to the vector field analysis signals.
It is quite sure that the results will be very useful in future works on different areas. A pair of Clifford-Fourier transforms is defined in the framework of Clifford analysis, as operator exponentials with a Clifford algebra-valued kernel. It is a genuine Clifford analysis construct, which is shown to be a refinement of the classical multi-dimensional Fourier transform. An adequate operational calculus is developed. | 1 |
As well known the classical uni-dimensional Fourier transform is a very powerful tool which is very useful in Analysis as well as in many other areas of mathematics. It is also used in many practical applications like information storage procedures, for instance.
There have been several generalizations of the classical Fourier transform to higher dimensions like the hypercomplex Fourier or the Quaternionic Fourier transform which were introduced by Bülow and Sommer. One type of these generalizations is formed by the Clifford Fourier transforms, which are based on Clifford algebra on \(\mathbb R^m\) and Clifford analysis, whose basic notions are briefly and neatly introduced in the paper under review. One of the main difficulties of the study of these generalizations is the non-commutativity of Clifford Algebra and finding a kernel that allow one to work with the transform to extend the results of the uni-dimensional classical Fourier Transform.
The authors of the present paper in [``The Clifford-Fourier transform'', J. Fourier Anal. Appl. 11, No. 6, 669--681 (2005; Zbl 1122.42015)] introduced one of these types of Clifford transforms. This transform can be defined by
\[
(\mathcal F f)(y)={1\over 2\pi} \int_{\mathbb R^2} e^{x \wedge y} f(x)\,dA(x),
\]
where \(f\) is an integrable complex function on \(\mathbb R^2\). Originally, it was defined by decomposing the uni-dimensional Fourier transform a an exponential of a scalar operator involving the Laplacian. All these facts are clearly presented in the paper under review. Indeed, the concepts are motivated and kindly introduced so that even someone who is not familiar with the subject can read the paper. Furthermore, a deep study of their Clifford Fourier transform is done. In particular, thanks to the knowledge of the kernel of the transform, they are able to show that the classical results, like the inversion and the convolution theorem, extend to their Clifford Fourier transform. Also, they study other properties like relating their transform to the standard tensorial Fourier transform. The authors thus propose their generalization for higher dimensions. They compute Clifford Fourier transform of the characteristic function and give some application to the vector field analysis signals.
It is quite sure that the results will be very useful in future works on different areas. Two-dimensional finite element analysis has been used to find load-transfer relationships for translation of an infinitely long pile through undrained soil for a variety of soil-constitutive models. It has been shown that these load-transfer curves can be used as p-y curves in the analysis of single piles undergoing lateral pile head loading in undrained soils with nonlinear stress-strain laws. Lateral pile response deduced from 2-D analysis input to the subgrade reaction method has been compared to the behaviour of a single pile analysed using three-dimensional finite element analysis. Good agreement between the two methods for nonlinear soils suggests that the 2-D analysis may form a useful design method for calculation of \(p-y\) curves. | 0 |
This is the last paper of the present series on collectivity and geometry [for the previous papers, see Part V, ibid. 28, 2223-2240 (1987; Zbl 0633.22007)]. In this paper the authors give a systematic description of their symplectic model for the collective motions of an n-body system in a d-dimensional space, where d is an arbitrary but finite cardinal number. They determine all the maximal subalgebras of sp(2d,R) that contain the generators of the orthogonal subalgebra o(d): these are u(d), sp(2,R)\(\oplus o(d)\) and cm(d) (``cm'' here stands for ``collective motion'') for \(d\geq 3\), but there are three more subalgebras for \(d=2\). The authors discuss the spectra of the Hamiltonian for the pure many-body systems associated with the three maximal subalgebras and introduce monomial basis states for the irreducible representations of sp(6,R) in the positive discrete series: this enables them to discuss the spectra and shapes of transitional systems. In the final section they give a brief account of all the conclusions that can be reached from the work described in the present series of papers. This is the fifth paper in a series [\textit{M. Moshinsky}, J. Math. Phys. 25, 1555-1564 (1984; Zbl 0553.22006); \textit{E. Chacón}, \textit{P. Hess} and \textit{M. Moshinsky}, ibid. 25, 1565-1576 (1984; Zbl 0553.22007); \textit{O. Castaños}, \textit{E. Chacón} and \textit{M. Moshinsky}, ibid. 25, 2815- 2825 (1984; Zbl 0585.22016); \textit{M. Moshinsky}, \textit{M. Nicolescu} and \textit{R. T. Sharp}, ibid. 26, 2995-2998 (1985; Zbl 0585.22017)] which studies the application of symplectic groups to nuclear structure. In the present paper a study based on the Lie algebra \({\mathfrak sp}(4,R)\) leads to a collective model in a two-dimensional space which, though somewhat remote from the correct description of nuclear structure, gives some considerable insight into the more realistic collective nuclear models such as the interacting boson approximation which is basd on \({\mathfrak u}(6)\) and into the symplectic model based on \({\mathfrak sp}(6,R)\). | 1 |
This is the last paper of the present series on collectivity and geometry [for the previous papers, see Part V, ibid. 28, 2223-2240 (1987; Zbl 0633.22007)]. In this paper the authors give a systematic description of their symplectic model for the collective motions of an n-body system in a d-dimensional space, where d is an arbitrary but finite cardinal number. They determine all the maximal subalgebras of sp(2d,R) that contain the generators of the orthogonal subalgebra o(d): these are u(d), sp(2,R)\(\oplus o(d)\) and cm(d) (``cm'' here stands for ``collective motion'') for \(d\geq 3\), but there are three more subalgebras for \(d=2\). The authors discuss the spectra of the Hamiltonian for the pure many-body systems associated with the three maximal subalgebras and introduce monomial basis states for the irreducible representations of sp(6,R) in the positive discrete series: this enables them to discuss the spectra and shapes of transitional systems. In the final section they give a brief account of all the conclusions that can be reached from the work described in the present series of papers. For non-anticipative functionals, differentiable in Chitashvili's sense, the Itô formula for cadlag semimartingales is proved. Relations between different notions of functional derivatives are established. | 0 |
Let \(S\) and \(T\) be regular semigroups, \(T\) acting on \(S\) by endomorphisms on the left. If \(S\) or \(T\) is completely simple then, within the semidirect product \(S*T\), the set of all regular elements \(\text{Reg}(S*T)\) forms a subsemigroup. This gives rise to a partial binary operation \(*_r\) on the lattice of e-varieties of regular semigroups [e-pseudovarieties of finite regular semigroups] where \({\mathcal U}*_r{\mathcal V}\) is the e-variety [e-pseudovariety] generated by the class \(\{\text{Reg}(S*T)\mid S\in{\mathcal U},\;T\in{\mathcal V}\}\) so that \({\mathcal U}*_r{\mathcal V}\) is defined provided that either \(\mathcal U\) or \(\mathcal V\) consists entirely of completely simple semigroups [see \textit{P. R. Jones} and \textit{P. G. Trotter}, Trans. Am. Math. Soc. 349, No. 11, 4265-4310 (1997; Zbl 0892.20037)].
In the present paper associativity of the partial binary operation \(*_r\) is investigated. Let \(\mathcal{U,V,W}\) be e-varieties, two of which are completely simple. In the main result of the paper it is shown that \({\mathcal U}*_r({\mathcal V}*_r{\mathcal W})=({\mathcal U}*_r{\mathcal V})*_r{\mathcal W}\) holds provided that
\[
{\mathcal V}*_r{\mathcal W}=\bigcup(HS)^nP\{\text{Reg}(S*T)\mid S\in{\mathcal V},\;T\in{\mathcal W}\}
\]
where \(P,S,H\) denote the class operators of forming, respectively, all direct products, all regular subsemigroups and all morphic images of all members of a given class of regular semigroups. It follows that associativity holds, for example if \({\mathcal V}*_r{\mathcal W}\) is contained in the class of all locally inverse semigroups or the class of all \(E\)-solid semigroups. No example of a triple \(\mathcal{U,V,W}\) is known where associativity fails. Moreover, for e-pseudovarieties of finite regular semigroups, associativity holds whenever the product is defined. The semidirect product \(S*T\) of two regular semigroups \(S\) and \(T\) in general neither is a regular semigroup nor is the set \(\text{Reg}(S*T)\) of all regular elements a subsemigroup of \(S*T\). However, if at least one of the two semigroups is completely simple then the set \(\text{Reg}(S*T)\) is indeed a (regular) subsemigroup of \(S*T\). This leads to a definition of a version of a semidirect product which is appropriate for the study of regular semigroups and which is defined whenever one of the two factors is completely simple. The definition naturally extends to the level of e-varieties by putting \({\mathbf U}*{\mathbf V}\) to be the e-variety generated by the class \(\{\text{Reg}(S*T)\mid S\in{\mathbf U},\;T\in{\mathbf V}\}\), and again, this definition is meaningful as soon as either \(\mathbf U\) or \(\mathbf V\) is contained in the e-variety of all completely simple semigroups. The study of this partial operation on the lattice of all e-varieties of regular semigroups is the main subject of this important paper. A lot of decomposition results for well-known e-varieties are given, for example \(L{\mathbf I}={\mathbf I}*\mathbf{RZ}\) or \(\mathbf{ES=CR*G}\) where \(\mathbf I=\) inverse semigroups, \(L\mathbf I=\) locally inverse semigroups, \(\mathbf{RZ}=\) right zero semigroups, \(\mathbf G=\) groups. In addition, most of the given decomposition results lead to models of the bifree objects (termed e-free objects in this paper) in the decomposed e-varieties (in fact, the paper provides a uniform method to construct bifree objects in e-varieties of the form \(\mathbf{U*V}\), this method is applicable whenever the bifree objects exist). | 1 |
Let \(S\) and \(T\) be regular semigroups, \(T\) acting on \(S\) by endomorphisms on the left. If \(S\) or \(T\) is completely simple then, within the semidirect product \(S*T\), the set of all regular elements \(\text{Reg}(S*T)\) forms a subsemigroup. This gives rise to a partial binary operation \(*_r\) on the lattice of e-varieties of regular semigroups [e-pseudovarieties of finite regular semigroups] where \({\mathcal U}*_r{\mathcal V}\) is the e-variety [e-pseudovariety] generated by the class \(\{\text{Reg}(S*T)\mid S\in{\mathcal U},\;T\in{\mathcal V}\}\) so that \({\mathcal U}*_r{\mathcal V}\) is defined provided that either \(\mathcal U\) or \(\mathcal V\) consists entirely of completely simple semigroups [see \textit{P. R. Jones} and \textit{P. G. Trotter}, Trans. Am. Math. Soc. 349, No. 11, 4265-4310 (1997; Zbl 0892.20037)].
In the present paper associativity of the partial binary operation \(*_r\) is investigated. Let \(\mathcal{U,V,W}\) be e-varieties, two of which are completely simple. In the main result of the paper it is shown that \({\mathcal U}*_r({\mathcal V}*_r{\mathcal W})=({\mathcal U}*_r{\mathcal V})*_r{\mathcal W}\) holds provided that
\[
{\mathcal V}*_r{\mathcal W}=\bigcup(HS)^nP\{\text{Reg}(S*T)\mid S\in{\mathcal V},\;T\in{\mathcal W}\}
\]
where \(P,S,H\) denote the class operators of forming, respectively, all direct products, all regular subsemigroups and all morphic images of all members of a given class of regular semigroups. It follows that associativity holds, for example if \({\mathcal V}*_r{\mathcal W}\) is contained in the class of all locally inverse semigroups or the class of all \(E\)-solid semigroups. No example of a triple \(\mathcal{U,V,W}\) is known where associativity fails. Moreover, for e-pseudovarieties of finite regular semigroups, associativity holds whenever the product is defined. No review copy delivered. | 0 |
``We investigate the curvature of the so-called diagonal lift from an affine manifold to the linear frame bundle \(LM\). This is an affine analogue (but not a direct generalization) of the Sasaki-Mok metric on \(LM\) investigated by \textit{L. A. Cordero} and \textit{M. de León} in [J. Math. Pures Appl., IX. Sér. 65, 81--91 (1986; Zbl 0542.53014)].The Sasaki-Mok metric is constructed over a Riemannian manifold as base manifold''.
Very roughly speaking the main result is as follows. Let \(\nabla\) be a symmetric affine connection of a manifold \(M\) and \(\mathbf{g}\) be the diagonal lift of \(\nabla\) to its linear frame bundle \(LM\). Then \((LM, \mathbf{g})\) is of constant scalar curvature if and only if the base manifold \((M, \nabla)\) is flat, and hence \((LM, \mathbf{g})\) is flat. Let (M,G) be a Riemannian manifold, FM the frame bundle of M and \(G^ D\) the diagonal lift of G to FM with respect to the Levi-Civita connection of G [same authors, Rend. Circ. Mat. Palermo, II. Ser. 32, 236-271 (1983; Zbl 0524.53020)]; then \(G^ D\) actually is the Riemannian metric firstly considered by \textit{K. P. Mok} [J. Reine Angew. Math. 302, 16-31 (1978; Zbl 0378.53016)]. In this paper, the Levi-Civita connection of \(G^ D\) and its curvature tensor are computed, obtaining formulae for them similar to those given by \textit{O. Kowalski} [J. Reine Angew. Math. 250, 124-129 (1971; Zbl 0222.53044)] for the tangent bundle TM endowed with the Sasaki metric. Using these formulae the sectional, Ricci and scalar curvatures of \((FM,G^ D)\) are computed and related with the corresponding curvatures of (M,G); as a consequence some results on the differential geometry of \((FM,G^ D)\) are obtained, as for instance: Theorem: If \((FM,G^ D)\) is an Einstein manifold, then (M,G) is flat. | 1 |
``We investigate the curvature of the so-called diagonal lift from an affine manifold to the linear frame bundle \(LM\). This is an affine analogue (but not a direct generalization) of the Sasaki-Mok metric on \(LM\) investigated by \textit{L. A. Cordero} and \textit{M. de León} in [J. Math. Pures Appl., IX. Sér. 65, 81--91 (1986; Zbl 0542.53014)].The Sasaki-Mok metric is constructed over a Riemannian manifold as base manifold''.
Very roughly speaking the main result is as follows. Let \(\nabla\) be a symmetric affine connection of a manifold \(M\) and \(\mathbf{g}\) be the diagonal lift of \(\nabla\) to its linear frame bundle \(LM\). Then \((LM, \mathbf{g})\) is of constant scalar curvature if and only if the base manifold \((M, \nabla)\) is flat, and hence \((LM, \mathbf{g})\) is flat. We consider a general model of the form
\[
Y_t=g(X_t,X_{t-1},\dots),\quad t\in\mathbb Z
\]
where \((X_t,Y_t),t\in\mathbb Z\) is a strictly stationary process. We construct a consistent estimator of \(g\) and apply the obtained result to the estimation of Volterra series. | 0 |
The authors of this interesting paper investigate the statistical properties of the progress towards the minimum for a class of stochastic algorithms. When a deterministic algorithm for finding the minimum of a function on a set is employed, it may reach a local minimum and remain there forever after. Restarting repeatedly and independently by a random choice of a starting point when the algorithm reaches a settling point engenders a probability of \(\lambda^n/s\) of not having seen the goal state by the \(n\)th epoch.
\textit{R. Shonkwiler} and \textit{E. Van Vleck} [J. Complexity 10, No. 1, 64-95 (1994; Zbl 0798.90124)] studied restarting from the perspective of its consequence on first hitting times. The present results extend those of Shonkwiler and Van Vleck and introduce on-the-fly estimation of the rate of convergence of the algorithm and prove its asymptotic normality. The elements of the class of stochastic algorithms treated here are generated by randomly restarting a deterministic iterative scheme when it reaches a settling point. Given a real valued function \(C\) defined on a topological domain \(\Omega \) with neighborhood system \(N\), restart methods for finding some point \(x^{\ast }\in opt (C)=G\) are analyzed. The set of global minimizers of \(C\) over \(\Omega \), where \(x^{\ast }\in G\) entails \(C(x^{\ast })\leq C(x)\), \(x\in \Omega \) is analyzed as well.
In case \(C\) is sufficiently smooth, there are powerful numerical methods for finding local minimizers. Assume \(M\) is the union of all the basins whose settling points are global minima. Then the number \(T=\min \{t:X(t)\in M\}\) (the first hitting time of \(M\) by the search process) would be of primary interest. The main result is that if \(\mu \) places positive mass at all points of the finite set \(\Omega \), then there are \(\lambda \in (0,1)\), \(\delta \in (0,\lambda)\), and a constant \(s>1\) such that for any \(\varepsilon >0\) there exists \((\delta +\varepsilon)^{-n}|P[T>n]-\lambda^n/s|\to 0\) as \(n\to \infty \). It turns out that \(\lambda \) is the unique positive solution to the equation \(\lambda^{-1}\phi_{H|N}(\lambda^{-1})=(1-\theta_0)^{-1}\), where \(\phi_{H|N}\) is the probability generating function of the random time to first settling point given that the starting state is one which leads to a non-global extremum.
Results on expectation of time to hit the goal are presented as well. The consequences of independent identical processing, and how to estimate the parameter \(\lambda \) on the fly are also discussed. We introduce the notion of expected hitting time to a goal as a measure of the convergence rate of a Monte Carlo optimization method. The techniques developed apply to simulated annealing, genetic algorithms, and other stochastic search schemes. The expected hitting time can itself be calculated from the more fundamental complementary hitting time distribution (CHTD) which completely characterizes a Monte Carlo method. The CHTD is asymptotically a geometric series, \((1/s)/(1- \lambda)\), characterized by two parameters, \(s\), \(\lambda\), related to the search process in a simple way. The main utility of the CHTD is in comparing Monte Carlo algorithms. In particular, we show that independent, identical Monte Carlo algorithms run in parallel, IIP parallelism, and exhibit superlinear speedup. We give conditions under which this occurs and note that equally likely search is linearly sped up. Further, we observe that a serial Monte Carlo search can have an infinite expected hitting time, but the same algorithm when parallelized can have a finite expected hitting time. One consequence of the observed superlinear speedup is an improved uniprocessor algorithm by the technique of in-code parallelism. | 1 |
The authors of this interesting paper investigate the statistical properties of the progress towards the minimum for a class of stochastic algorithms. When a deterministic algorithm for finding the minimum of a function on a set is employed, it may reach a local minimum and remain there forever after. Restarting repeatedly and independently by a random choice of a starting point when the algorithm reaches a settling point engenders a probability of \(\lambda^n/s\) of not having seen the goal state by the \(n\)th epoch.
\textit{R. Shonkwiler} and \textit{E. Van Vleck} [J. Complexity 10, No. 1, 64-95 (1994; Zbl 0798.90124)] studied restarting from the perspective of its consequence on first hitting times. The present results extend those of Shonkwiler and Van Vleck and introduce on-the-fly estimation of the rate of convergence of the algorithm and prove its asymptotic normality. The elements of the class of stochastic algorithms treated here are generated by randomly restarting a deterministic iterative scheme when it reaches a settling point. Given a real valued function \(C\) defined on a topological domain \(\Omega \) with neighborhood system \(N\), restart methods for finding some point \(x^{\ast }\in opt (C)=G\) are analyzed. The set of global minimizers of \(C\) over \(\Omega \), where \(x^{\ast }\in G\) entails \(C(x^{\ast })\leq C(x)\), \(x\in \Omega \) is analyzed as well.
In case \(C\) is sufficiently smooth, there are powerful numerical methods for finding local minimizers. Assume \(M\) is the union of all the basins whose settling points are global minima. Then the number \(T=\min \{t:X(t)\in M\}\) (the first hitting time of \(M\) by the search process) would be of primary interest. The main result is that if \(\mu \) places positive mass at all points of the finite set \(\Omega \), then there are \(\lambda \in (0,1)\), \(\delta \in (0,\lambda)\), and a constant \(s>1\) such that for any \(\varepsilon >0\) there exists \((\delta +\varepsilon)^{-n}|P[T>n]-\lambda^n/s|\to 0\) as \(n\to \infty \). It turns out that \(\lambda \) is the unique positive solution to the equation \(\lambda^{-1}\phi_{H|N}(\lambda^{-1})=(1-\theta_0)^{-1}\), where \(\phi_{H|N}\) is the probability generating function of the random time to first settling point given that the starting state is one which leads to a non-global extremum.
Results on expectation of time to hit the goal are presented as well. The consequences of independent identical processing, and how to estimate the parameter \(\lambda \) on the fly are also discussed. We study the orbital stability of single-spike semiclassical standing waves of a nonhomogeneous in space nonlinear Schrödinger-Poisson equation. When the nonlinearity is subcritical or supercritical we prove that the nonlocal Poisson-term does not influence the stability of standing waves, whereas in the critical case it may create instability if its value at the concentration point of the spike is too large. The proofs are based on the study of the spectral properties of a linearized operator and on the analysis of a slope condition. Our main tools are perturbation methods and asymptotic expansion formulas. | 0 |
The authors study the topological space \({\mathcal C}(H^\infty)\) (endowed with the operator-norm topology) of composition operators \(C_\varphi\) on the space \(H^\infty\) of bounded analytic functions on the unit disk \(\mathbb{D}\). They answer a question of \textit{B. MacCluer, Sh. Ohno} and \textit{R.-H. Zhao} [Integral Equations Oper. Theory 40, No. 4, 481-494 (2001; Zbl 1062.47511)] by showing that \(C_\varphi\) is isolated in \({\mathcal C}(H^\infty)\) if and only if \(C_\varphi\) is essentially isolated. Recall that the essential semi-norm on \({\mathcal C}(H^\infty)\) is given by
\[
\text{\(\|C_\varphi\|_e:=\inf\{\|C_\varphi-K\|: K\) is compact on \(H^\infty\}\)}.
\]
The proof uses their newly introduced notion of asymptotically interpolating sequence, a subject interesting in its own right [see \textit{P. Gorkin} and \textit{R. Mortini}, J. Lond. Math. Soc., II. Ser. 67, No. 2, 481-498 (2003)].
It is also shown that \(C_\varphi\) is isolated in \({\mathcal C}(H^\infty)\) if and only if \(\int_0^{2\pi} \log(1-|\varphi|) d\theta=-\infty\), that is iff \(\varphi\) is an extreme point of the unit ball in \(H^\infty\). Components and isolated points of the topological space of composition operators on \(H^\infty\) in the uniform operator topology are characterized. Compact differences of two composition operators are also characterized. With the aid of these results, we show that a component in \({\mathcal C}(H^{\infty})\) is not in general the set of all composition operators that differ from the given one by a compact operator. | 1 |
The authors study the topological space \({\mathcal C}(H^\infty)\) (endowed with the operator-norm topology) of composition operators \(C_\varphi\) on the space \(H^\infty\) of bounded analytic functions on the unit disk \(\mathbb{D}\). They answer a question of \textit{B. MacCluer, Sh. Ohno} and \textit{R.-H. Zhao} [Integral Equations Oper. Theory 40, No. 4, 481-494 (2001; Zbl 1062.47511)] by showing that \(C_\varphi\) is isolated in \({\mathcal C}(H^\infty)\) if and only if \(C_\varphi\) is essentially isolated. Recall that the essential semi-norm on \({\mathcal C}(H^\infty)\) is given by
\[
\text{\(\|C_\varphi\|_e:=\inf\{\|C_\varphi-K\|: K\) is compact on \(H^\infty\}\)}.
\]
The proof uses their newly introduced notion of asymptotically interpolating sequence, a subject interesting in its own right [see \textit{P. Gorkin} and \textit{R. Mortini}, J. Lond. Math. Soc., II. Ser. 67, No. 2, 481-498 (2003)].
It is also shown that \(C_\varphi\) is isolated in \({\mathcal C}(H^\infty)\) if and only if \(\int_0^{2\pi} \log(1-|\varphi|) d\theta=-\infty\), that is iff \(\varphi\) is an extreme point of the unit ball in \(H^\infty\). Some discrete analogue of Poincaré-type integral inequalities involving many independent variables are established. These in turn can be used to serve as generators of other interesting discrete inequalities. | 0 |
Let R be a commutative ring and M a finitely generated \((= f.g.)\) multiplication R-module, i.e., every submodule N of M has the form JM for some ideal J of R. Putting \(D=AnnAnn(M)\) and \(M^*=\) the dual of M, the authors show:
Theorem. Let M be a f.g. multiplication module, and assume that D is a f.g. projective (or flat) ideal, then \(M^*\) is a f.g. projective (resp. flat) module.
Corollary. With assumptions as above, if \(Ann(M)=Ann(D)\), then M is projective.
[See also the following review Zbl 0685.13004.] [For part I see the preceding review.]
Let R be a commutative ring and M a finitely generated \((= f.g.)\) multiplication R-module, i.e., every submodule N of M has the form JM for some ideal J of R. Putting \(D=AnnAnn(M)\) and \(M^*=\) the dual of M, the author shows:
Theorem. Let M be a f.g. multiplication module, and assume that D is a projective (or flat) ideal, then \(M^*\) is a projective (resp. flat) module.
Theorem. Let M be a f.g. multiplication module. Then \(M^*\) is a multiplication module if and only if D is a multiplication ideal. | 1 |
Let R be a commutative ring and M a finitely generated \((= f.g.)\) multiplication R-module, i.e., every submodule N of M has the form JM for some ideal J of R. Putting \(D=AnnAnn(M)\) and \(M^*=\) the dual of M, the authors show:
Theorem. Let M be a f.g. multiplication module, and assume that D is a f.g. projective (or flat) ideal, then \(M^*\) is a f.g. projective (resp. flat) module.
Corollary. With assumptions as above, if \(Ann(M)=Ann(D)\), then M is projective.
[See also the following review Zbl 0685.13004.] Gravitational holography is argued to render the cosmological constant stable against divergent quantum corrections, thus providing a technically natural solution to the cosmological constant problem. Evidence for quantum stability of the cosmological constant is illustrated in a number of examples, including bulk descriptions in terms of delocalized degrees of freedom, boundary screen descriptions on stretched horizons, and nonsupersymmetric conformal field theories as dual descriptions of anti-de Sitter space. In an expanding universe, holographic quantum contributions to the stress-energy tensor are argued to be at most of order the energy density of the dominant matter component. | 0 |
The authors introduce and study the Fitzpatrick transform of a monotone bifunction. This generalizes the notion of the Fitzpatrick function which is of central importance in monotone operator theory; see, e.g., \textit{S. Simons}'s monograph [From Hahn--Banach to monotonicity. 2nd expanded ed. Lecture Notes in Mathematics 1693. Berlin: Springer (2008; Zbl 1131.47050)]. Alas, no example is offered to illustrate the results in this paper. These notes are somewhere between a sequel to and a new edition of [``Minimax monotonicity'' (Lect.\ Notes Math. 1693, Berlin: Springer) (1998; Zbl 0922.47047)].
As in [op.\,cit.], the essential idea is to reduce questions on monotone multifunctions to questions on convex functions. In [op.\,cit.], this was achieved using a ``big convexification'' of the graph of the multifunction and the ``minimax technique'' for proving the existence of linear functionals satisfying certain conditions. The ``big convexification'' is a very abstract concept, and the analysis is quite heavy in computation. The Fitzpatrick function gives another, more concrete, way of associating a convex functions with a monotone multifunction. The problem is that many of the questions on convex functions that one obtains require an analysis of the special properties of convex functions on the product of a Banach space with its dual, which is exactly what we do in these notes. It is also worth noting that the minimax theorem is hardly used here.
We envision that these notes could be used for four different possible courses/seminars:
\(\bullet\) An introductory course in functional analysis which would, at the same time, touch on minimax theorems and give a grounding in convex Lagrange multiplier theory and the main theorems in convex analysis.
\(\bullet\) A course in which results on monotonicity on general Banach spaces are established using symmetrically self-dual spaces and Fitzpatrick functions.
\(\bullet\) A course in which results on monotonicity on reflexive Banach spaces are S established using symmetrically self-dual spaces and Fitzpatrick functions.
\(\bullet\) A seminar in which the the more technical properties of maximal monotonicity on general Banach spaces that have been established since 1997 are discussed. | 1 |
The authors introduce and study the Fitzpatrick transform of a monotone bifunction. This generalizes the notion of the Fitzpatrick function which is of central importance in monotone operator theory; see, e.g., \textit{S. Simons}'s monograph [From Hahn--Banach to monotonicity. 2nd expanded ed. Lecture Notes in Mathematics 1693. Berlin: Springer (2008; Zbl 1131.47050)]. Alas, no example is offered to illustrate the results in this paper. The paper outlines a project in the philosophy of mathematics based on a proposed
view of the nature of mathematical reasoning. It also contains a brief evaluative overview of
the discipline and some historical observations; here it points out and illustrates the division
between the philosophical dimension, where questions of realism and the status of mathematics
are treated, and the more descriptive and looser dimension of epistemic efficiency, which has to
do with ways of organizing the mathematical material. The paper's concern is with the first.
The grand tradition in the philosophy of mathematics goes back to the foundational debates at
the end of the 19th and the first decades of the 20th century. Logicism went together with a
realistic view of actual infinities; rejection of, or skepticism about actual infinities derived from
conceptions that were Kantian in spirit. Yet questions about the nature of mathematical reasoning should be distinguished from questions about realism (the extent of objective knowledge--independent mathematical truth). Logicism is now dead. Recent attempts to revive it are based
on a redefinition of ``logic'', which exploits the flexibility of the concept; they yield no interest-
ing insight into the nature of mathematics. A conception of mathematical reasoning, broadly
speaking along Kantian lines, need not imply antirealism and can be pursued and investigated,
leaving questions of realism open. Using some concrete examples of nonformal mathematical
proofs, the paper proposes that mathematics is the study of forms of organization -- a concept
that should be taken as primitive, rather than interpreted in terms of set-theoretic structures.
For set theory itself is a study of a particular form of organization, albeit one that provides a
modeling for the other known mathematical systems. In a nutshell: ``We come to know mathematical truths through becoming aware of the properties of some of the organizational forms that
underlie our world. This is possible, due to a capacity we have: to reflect on some of our own
practices and the ways of organizing our world, and to realize what they imply. In this respect
all mathematical knowledge is meta-knowledge; mathematics is a meta-activity par excellence.''
This of course requires analysis and development, hence the project. The paper also discusses
briefly the axiomatic method and formalized proofs in light of the proposed view. | 0 |
The paper presents a comprehensive survey and some new results concerning two questions on end extensions of models of fragments of PA. The first, due to Paris, asks whether every countable model of \(\text{B}\Sigma_1\) has a proper end extension to a model of \(\text{I}\Delta_0\); the second, due to Clote, asks whether every countable model of \(\text{B}\Sigma_n\), \(n\geq 2\), has a proper \(\Sigma_n\)-elementary end extension to a model of \(\text{B}\Sigma_{n-1}\). Concerning the first question, \textit{A. Wilkie} and \textit{J. Paris} proved in [``On the existence of end extensions of models of bounded induction'', in: J. E. Fenstad et al. (eds.), Logic, methodology and philosophy of science VIII, Proc. 8th Int. Congr., Moscow/USSR 1987, Stud. Logic Found. Math. 126, 143--161 (1989; Zbl 0695.03019)] that the answer is affirmative if the model is \(\Delta_0\)-full. Cornaros and Dimitracopoulos analyze this notion and relate it to other notions of weak saturation. They introduce a notion of \(\Gamma\)-1-fullness, where \(\Gamma\) is a class of formulas, and they prove that if \(\Gamma\supseteq \text{I}\Delta_0\) is a recursive set of sentences and \(M\) is a countable model of \(\text{B}\Sigma_1+\Pi_1(\Gamma)\) and there is a sentence \(\lambda\) such that \(M\models\lnot\lambda\) and \(M\) is \((\Gamma+\lambda)\)-1-full, then there exists \(K\models\Gamma\) which is a proper \(\Sigma_1\)-elementary end extension of \(M\). They also prove that if there is a \(K\) as above and \(\Gamma\supseteq \text{I}\Sigma_1\), then \(M\) is \(\Gamma\)-1-full. Concerning Clote's question, Cornaros and Dimitracopoulos prove that for \(n\geq 2\) every countable model of \(\text{B}\Sigma_n\) has a proper \(\Sigma_n\)-elementary end extension to a model of parameter-free collection scheme for \(\Sigma_{n-1}\) formulas. They also modify a proof from the above mentioned paper of Wilkie and Paris to show that for every \(n\geq 2\) there exists a countable \(M\models \text{B}\Sigma_n\) for which there is no proper \(\Sigma_{n-1}\)-elementary end extension to a model of \(\text{I}\Sigma_{n-1}\). [For the entire collection see Zbl 0676.00003.]
The authors consider the problem of determining when a countable model of bounded induction (that is, Peano Arithmetic with induction restricted to bounded formulas) has a proper end extension to a model of bounded induction. A necessary condition is clearly that the given model should satisfy \(\Sigma_ 1\)-collection. It is still unknown whether this is sufficient. However, if the bounded quantifier hierarchy collapses provably in bounded induction then it is not. The authors give further results under this assumption as well as providing a condition on models (called fullness) that guarantees the existence of end extensions. | 1 |
The paper presents a comprehensive survey and some new results concerning two questions on end extensions of models of fragments of PA. The first, due to Paris, asks whether every countable model of \(\text{B}\Sigma_1\) has a proper end extension to a model of \(\text{I}\Delta_0\); the second, due to Clote, asks whether every countable model of \(\text{B}\Sigma_n\), \(n\geq 2\), has a proper \(\Sigma_n\)-elementary end extension to a model of \(\text{B}\Sigma_{n-1}\). Concerning the first question, \textit{A. Wilkie} and \textit{J. Paris} proved in [``On the existence of end extensions of models of bounded induction'', in: J. E. Fenstad et al. (eds.), Logic, methodology and philosophy of science VIII, Proc. 8th Int. Congr., Moscow/USSR 1987, Stud. Logic Found. Math. 126, 143--161 (1989; Zbl 0695.03019)] that the answer is affirmative if the model is \(\Delta_0\)-full. Cornaros and Dimitracopoulos analyze this notion and relate it to other notions of weak saturation. They introduce a notion of \(\Gamma\)-1-fullness, where \(\Gamma\) is a class of formulas, and they prove that if \(\Gamma\supseteq \text{I}\Delta_0\) is a recursive set of sentences and \(M\) is a countable model of \(\text{B}\Sigma_1+\Pi_1(\Gamma)\) and there is a sentence \(\lambda\) such that \(M\models\lnot\lambda\) and \(M\) is \((\Gamma+\lambda)\)-1-full, then there exists \(K\models\Gamma\) which is a proper \(\Sigma_1\)-elementary end extension of \(M\). They also prove that if there is a \(K\) as above and \(\Gamma\supseteq \text{I}\Sigma_1\), then \(M\) is \(\Gamma\)-1-full. Concerning Clote's question, Cornaros and Dimitracopoulos prove that for \(n\geq 2\) every countable model of \(\text{B}\Sigma_n\) has a proper \(\Sigma_n\)-elementary end extension to a model of parameter-free collection scheme for \(\Sigma_{n-1}\) formulas. They also modify a proof from the above mentioned paper of Wilkie and Paris to show that for every \(n\geq 2\) there exists a countable \(M\models \text{B}\Sigma_n\) for which there is no proper \(\Sigma_{n-1}\)-elementary end extension to a model of \(\text{I}\Sigma_{n-1}\). We define Wiener amalgam spaces of (quasi)analytic ultradistributions whose local components belong to a general class of translation and modulation invariant Banach spaces of ultradistributions and their global components are either weighted \(L^p\) or weighted \(\mathcal{C}_0\) spaces. We provide a discrete characterisation via so called uniformly concentrated partitions of unity. Finally, we study the complex interpolation method and we identify the strong duals for most of these Wiener amalgam spaces. | 0 |
Let \(C\) be a projective, smooth quartic plane curve defined over a closed finite field \(k\) of characteristic zero. Let \(\ell\) be a line and \(\pi_P\) the projection of \(C\) to \(\ell\) centered at a point \(P\in C\). Moreover let \(C_P\) be the nonsingular model corresponding to the Galois closure of the extension \(k(C)| k(\ell)\) corresponding to \(\pi_P\); such a curve is called the Galois closure of \(C\) with respect to \(\pi_P\). The subject of this paper is concerning with the computation of the genus \(g_P\) of \(C_P\).
This problem was considered by \textit{K. Miura} and \textit{H. Yoshihara} [J. Algebra 226, 283--294 (2000; Zbl 0983.11067)] who showed that \(g_P\in \{3, 6,7,8,9,10\}\). Furthermore they showed that the cases 3,8,9,10 occur but the cases of the existence of quartics with \(g_P=6,7\) remained an open problem for several years. In this paper Watanabe prove that in fact the cases 6 and 7 also occur. There is a relate result which ask for the set of genus \(g_P\) for points \(P\) in a fixed quartic. It is shown that this set is different from \(\{3,6,7,8,9,10\}\). The approach of this paper is based on the Weierstrass point theory of quartic plane curves. Let \(K\) be the rational function field of a smooth plane curve \(C\) of degree \( d\) (\(d\geq 2)\) defined over an algebraic number field \(k\) of characteristic zero. If \(K_m\) denotes a maximal rational subfield of \(K\), the authors of the paper under review answer, in the case where \(C\) is a quartic curve, the following questions. (1) When is the extension \(K/K_m\) Galois? (2) Let \(L\) be the Galois closure of \(K/K_m\). What could we say about \(L\)? (3) What is the Galois group \(\text{Gal}(L/K_m)\)? The characterisation is dependent on the point \(P\), which is the center of the projection of the curve \(C\) to a line \(l\), and of the genus \(g(P)\) of the curve \(\grave{C}\), which corresponds to the field \(L\). | 1 |
Let \(C\) be a projective, smooth quartic plane curve defined over a closed finite field \(k\) of characteristic zero. Let \(\ell\) be a line and \(\pi_P\) the projection of \(C\) to \(\ell\) centered at a point \(P\in C\). Moreover let \(C_P\) be the nonsingular model corresponding to the Galois closure of the extension \(k(C)| k(\ell)\) corresponding to \(\pi_P\); such a curve is called the Galois closure of \(C\) with respect to \(\pi_P\). The subject of this paper is concerning with the computation of the genus \(g_P\) of \(C_P\).
This problem was considered by \textit{K. Miura} and \textit{H. Yoshihara} [J. Algebra 226, 283--294 (2000; Zbl 0983.11067)] who showed that \(g_P\in \{3, 6,7,8,9,10\}\). Furthermore they showed that the cases 3,8,9,10 occur but the cases of the existence of quartics with \(g_P=6,7\) remained an open problem for several years. In this paper Watanabe prove that in fact the cases 6 and 7 also occur. There is a relate result which ask for the set of genus \(g_P\) for points \(P\) in a fixed quartic. It is shown that this set is different from \(\{3,6,7,8,9,10\}\). The approach of this paper is based on the Weierstrass point theory of quartic plane curves. Für die ganze Funktion \(\sum\limits_{n=0}^\infty a_nz^n\) zeigt Verf. mittels einer schon früher (Sur certains invariants attachés aux fonctions analytiques, Mathematica, Cluj, 12 (1936), 164-179; F. d. M. \(62_{\text{II}}\)) dargelegten Methode auf direktem Wege die Invarianz von
\[
\begin{gathered} -\frac1\varrho=\varlimsup\frac{\log|a_n|}{n\log n},\quad \varrho_1=\varlimsup\frac{\varrho\log|a_n|+n\log n}{n\log_2n},\ldots,\\ \varrho_k=\varlimsup\frac {\varrho\log|a_n|+n\log n-\varrho_1 n\log_2n-\cdots-\varrho_{k-1}n\log_kn} {n\log_{k+1}n},\ldots \end{gathered}
\]
gegenüber analytischer Fortsetzung, was sonst mit Ungleichungen von \textit{Lindelöf} und \textit{Boutroux} bewiesen wird. | 0 |
The author intends to generalize a result of \textit{F. Halter-Koch} [J. Number Theory 34, 82--94 (1990; Zbl 0697.12007)], and provides lower bounds for class numbers \(h(d)\) of real quadratic fields \(\mathbb Q(\sqrt{d})\) of narrow Richaud-Degert type (i.e. \(d=a^ 2+r\), \(r=\pm 1,\pm 4)\) in terms of the divisor function \(\tau(x)\) which means the number of distinct positive divisors of \(x\). He proves for instance that if \(d=a^ 2+1\) \((a>1\) odd) then \(h(d)\geq 2\tau(a)-2\). In connection with the class number one problem, many results on lower bounds for the class number of real quadratic fields are recently obtained by using diophantine equation theory [the reviewer, Proc. Int. Conf. Katata/Jap., 125-137 (1986; Zbl 0612.12010), Adv. Stud. Pure Math. 13, 493-501 (1988; Zbl 0664.10002), \textit{R. A. Mollin}, Proc. Am. Math. Soc. 101, 439-444 (1987; Zbl 0632.12006)] and by using continued fraction theory [\textit{T. Azuhata}, Tokyo J. Math. 10, 259-270 (1987; Zbl 0659.12008), \textit{S. Louboutin}, J. Number Theory 30, No.2, 167-176 (1988; Zbl 0652.12002)], respectively.
In this paper, the author intends to generalize and improve these already known results. Namely, he considers generally the quadratic orders and obtains a general theorem on lower bounds for the class number, which is applicable directly for imaginary quadratic orders and is available for real quadratic orders if we have norms of reduced principal ideals, especially the continued fraction expansion of the basis element of the order. | 1 |
The author intends to generalize a result of \textit{F. Halter-Koch} [J. Number Theory 34, 82--94 (1990; Zbl 0697.12007)], and provides lower bounds for class numbers \(h(d)\) of real quadratic fields \(\mathbb Q(\sqrt{d})\) of narrow Richaud-Degert type (i.e. \(d=a^ 2+r\), \(r=\pm 1,\pm 4)\) in terms of the divisor function \(\tau(x)\) which means the number of distinct positive divisors of \(x\). He proves for instance that if \(d=a^ 2+1\) \((a>1\) odd) then \(h(d)\geq 2\tau(a)-2\). In this paper we recall some recent results about variational eigenvalues of the p-Laplacian, we show new applications and point out some open problems. We focus on the continuity properties of the eigenvalues under the \(\mathrm{gamma_p}\)-convergence of capacitary measures, which are needed to prove existence results for the minimization of nonlinear eigenvalues in the class of p-quasi open sets contained in a box under a measure constraint.
Finally, the new contribution of this paper is to show that these continuity results can be employed to prove existence of minimizers for nonlinear eigenvalues among measurable sets contained in a box and under a perimeterconstraint. | 0 |
A graph \(G\) is minimally \(1\)-factorable if \(G\) has at least one \(1\)-factorization and if every \(1\)-factor of \(G\) is contained in a unique \(1\)-factorization of \(G\). \textit{M. Funk} and \textit{D. Labbate} [Discrete Math. 216, 121-137 (2000; Zbl 0952.05055)] have shown that an \(r\)-regular bipartite minimally \(1\)-factorable graph exists if and only if \(r\leq 3\). In this paper the latter author continues his study of cubic bipartite minimally \(1\)-factorable graphs. He shows that such graphs must have an odd number of vertices in each of their two vertex sets. He then proves his main result, which is that if a cubic bipartite minimally \(1\)-factorable graph has girth \(4\) then the determinant of its biadjacency matrix is zero. He conjectures that for arbitrary girth this determinant is either zero or equal (up to sign) to the permanent of the biadjacency matrix. The famous Hall marriage theorem implies that every one-factor of an \(r\)-regular bipartite graph can be completed to a one-factorization, i.e. to a factorization of \(r\) one-factors. The authors study two classes of \(r\)-regular bipartite graphs. The first class is the class of minimally one-factorable graphs, i.e. graphs in which every one-factor belongs to precisely one one-factorization. It is proved that the Heawood graph is an instance of a minimally one-factorable graph. Also they show that minimally one-factorable \(r\)-regular bipartite graphs exist only for \(r\leq 3\). The authors construct a wide class \(\mathcal E\) of examples of minimally one-factorable cubic bipartite graphs containing the Heawood graph. The second class of graphs, the det-extremal graphs, is the class of bipartite graphs \(G\) with \(\det(G)=\text{per}(G)\). (\(\det(G)\) is the determinant and \(\text{per}(G)\) is the permanent of the adjacency matrix of \(G\).) In general \(\det(G)\leq \text{per}(G)\) and it is well known that \(\text{per}(G)\) equals the number of one-factors of \(G\). Using a numeric evaluation by computer the authors prove that the Heawood graph is the only cubic det-extremal graph with less than 26 vertices. Finally they show that disregarding the Heawood graph, the class \(\mathcal E\) contains a second instance of a det-extremal graph. | 1 |
A graph \(G\) is minimally \(1\)-factorable if \(G\) has at least one \(1\)-factorization and if every \(1\)-factor of \(G\) is contained in a unique \(1\)-factorization of \(G\). \textit{M. Funk} and \textit{D. Labbate} [Discrete Math. 216, 121-137 (2000; Zbl 0952.05055)] have shown that an \(r\)-regular bipartite minimally \(1\)-factorable graph exists if and only if \(r\leq 3\). In this paper the latter author continues his study of cubic bipartite minimally \(1\)-factorable graphs. He shows that such graphs must have an odd number of vertices in each of their two vertex sets. He then proves his main result, which is that if a cubic bipartite minimally \(1\)-factorable graph has girth \(4\) then the determinant of its biadjacency matrix is zero. He conjectures that for arbitrary girth this determinant is either zero or equal (up to sign) to the permanent of the biadjacency matrix. The management of computational resources is becoming a crucial aspect in new generation distributed computing systems like the Grid because of the decentralized, heterogeneous and autonomous nature of these resources. As such they cannot be managed by adopting a centralized approach, but more sophisticated computing methodologies are necessary. In this paper we propose to use software agent negotiation to select services necessary to compose Grid applications. In particular, we propose an automated negotiation mechanism to select the service providers that meet the requirements of service consumers on the provision of multiple interconnected services. The negotiation mechanism allows for the evaluation of dependent issues that are negotiated upon when multiple interconnected services are required, and it relies on an iterative process so to improve the possibility of reaching an agreement by letting both service consumers and providers to exchange more proposals and counter-proposals in order to accommodate to the dynamic and changing nature of Grid environments. | 0 |
The authors of this paper consider the resolvent estimate for the Laplacian in Euclidean spaces. Precisely, it is of the form
\[
\|(-\Delta -z)^{-1}f\|_{L^q(\mathbb{R}^d)}\leq C\|f\|_{L^p(\mathbb{R}^d)}\,,\qquad \forall\;z\in\mathbb{C}\backslash [0,\infty).
\]
They provide a complete characterization of sharp \(L^p\)-\(L^q\) resolvent estimates which could depend on \(z\). Additionally, they consider sharp resolvent estimates for the fractional Laplacians. Further, the obtain new results for the Bochner-Riesz operators of negative index. The main goal of this paper is to prove ``Sobolev inequalities'' for constant coefficient second order differential operators with bounds which do not depend on the lower order terms. More precisely, let Q(\(\xi)\) denote a non singular real quadratic form on \({\mathbb{R}}^ n\), \(n\geq 3\), which, for some \(2\leq j\leq n\), is given by
\[
Q(\xi) = =\xi^ 2_ 1-...-\xi^ 2_ j\quad +\quad \xi^ 2_{j+1}+...+\xi^ 2_ n.
\]
Then, if dual exponents p and p' enjoy the relationship \(1/p- 1/p'=2/n\), there is an absolute constant C such that whenever P(D) is a constant coefficient operator with complex coefficients and principal part Q(D) one has:
\[
\| u\|_{L^{p'}({\mathbb{R}}^ n)}\leq C\| P(D)u\|_{L^ p({\mathbb{R}}^ n)},\quad u\in H^{2,P}({\mathbb{R}}^ n).
\]
The main motivation behind proving these uniform Sobolev inequalities is that they imply certain local or global unique continuation theorems for operators P(D) as above. More specifically, they imply certain Carleman inequalities which in turn give unique continuation results for solutions to Schrödinger equations of the form \(P(D)u+Vu=0\), when certain conditions are placed on the solution u and the potential V. | 1 |
The authors of this paper consider the resolvent estimate for the Laplacian in Euclidean spaces. Precisely, it is of the form
\[
\|(-\Delta -z)^{-1}f\|_{L^q(\mathbb{R}^d)}\leq C\|f\|_{L^p(\mathbb{R}^d)}\,,\qquad \forall\;z\in\mathbb{C}\backslash [0,\infty).
\]
They provide a complete characterization of sharp \(L^p\)-\(L^q\) resolvent estimates which could depend on \(z\). Additionally, they consider sharp resolvent estimates for the fractional Laplacians. Further, the obtain new results for the Bochner-Riesz operators of negative index. It is shown that segment congruence can be defined by means of a positive existential definition in terms of segment inequality in several geometries over Archimedean ordered fields, and that it can be defined by means of a positive sentence in \(n\)-dimensional Euclidean geometry over arbitrary ordered fields. | 0 |
Let \(M\) be a finite-dimensional complex linear representation of a torus \(G\). The condition that the orbit dimension is at least \(k\) filters \(M\) by open sets \(M_{k}\). \(T^{*}M\) has a \(G\)-action and canonical action form that define a moment map whose zero set is the transverse cotangent bundle \(T_{G}^{*}M\) of \(M\). The authors show that the infinitesimal index in [the authors, ``The infinitesimal index'', Preprint (2010), \url{arXiv:1003.3525}] defines an isomorphism between the equivariant cohomology with compact supports of \(T_{G}^{*}M_{k}\) and a space of splines introduced in [the authors, Transform. Groups 15, No. 4, 751--773 (2010; Zbl 1223.58015)], where similar ideas were expressed in the language of equivariant \(K\)-theory and transversally elliptic operators. The authors plan to use their infinitesimal index to give explicit index formulas for transversally elliptic operators. This is the first in a series of papers of the authors on vector partition functions and the index theory of transversally elliptic operators. In this paper, only algebraic and combinatorial issues related to partition functions are discussed.
Let \(\Gamma\) be a lattice in a vector space \(V\) and \(X\) be a list of nonzero elements of \(\Gamma\) spanning \(V\). If \(X\) generates a pointed cone, the partition function \(\mathcal{P}_X(\gamma)\) counts the number of ways in which \(\gamma\in \Gamma\) can be written as a linear combination of elements in \(X\) with nonnegative integer coefficients.
A sublist \(Y\) of \(X\) is called a cocircuit if \(X\setminus Y\) does not span \(V\) and \(Y\) is minimal with this property. Given \(a\in \Gamma,\) the difference operator \(\nabla_a\) acts on functions by \(\nabla_a(f)(b):=f(b)-f(b-a).\) Given a list \(Y\), let \(\nabla_Y:=\prod_{a\in Y} \nabla_a.\)
\textit{W. Dahmen} and \textit{C. A. Micchelli} [Trans. Am. Math. Soc. 308, No.~2, 509--532 (1988; Zbl 0655.10013)] introduced the space \(DM(X)\) consisting of functions \(f\) on \(\Gamma\) satisfying \(\nabla_Y(f)=0\) for every cocircuit \(Y\). Every function in \(DM(X)\) is a quasi-polynomial, i.e., a function which coincides with a polynomial on each coset of some sublattice of finite index in \(\Gamma.\)
In this paper, a space \(\mathcal{F}(X)\) generating the space \(DM(X)\) is introduced. A subspace \(\underline{r}\) of \(V\) is called rational if it is the span of a sublist of \(X\). The space \(\mathcal{F}(X)\) consists of functions \(f\in \Gamma\) such that \(\nabla_{X\setminus \underline{r}}(f)\) is supported on \(\underline{r}\) for every rational \(\underline{r}\).
The main result of this paper is a localization formula for a function in \(\mathcal{F}(X)\). A tope is a connected component of the complement in \(V\) of the union of all proper rational subspaces. Given a tope \(\tau,\) a function \(f\in \mathcal{F}(X)\) can be written as a sum of a quasi-polynomial \(f^\tau\in DM(X)\) and of functions \(f_{\underline{r}}\in \mathcal{F}(X)\) supported outside the Minkowski difference \(\tau-B(X)\), where \(B(X)\) is the zonotope of \(X\), i.e., the set of linear combinations of elements in \(X\) with real coefficients in the interval \([0,1].\)
Since the partition function \(\mathcal{P}_X\) is contained in \(\mathcal{F}(X)\), the localization formula, together with a wall crossing formula, allows the authors to give a short proof of the result that the partition function \(\mathcal{P}_X\) is a quasi-polynomial on the Minkowski difference \(\mathfrak{c}-B(X)\), where \(\mathfrak{c}\) is a big cell, i.e., a connected component of the complement in \(V\) of the union of all cones spanned by all sublists of \(X\) which do not span \(V\).
This result was proved by Dahmen and Micchelli [loc. cit.] for topes and by \textit{A. Szenes} and \textit{M. Vergne} [Adv. Appl. Math. 30, No.~1--2, 295--342 (2003; Zbl 1067.52014)] for big cells. | 1 |
Let \(M\) be a finite-dimensional complex linear representation of a torus \(G\). The condition that the orbit dimension is at least \(k\) filters \(M\) by open sets \(M_{k}\). \(T^{*}M\) has a \(G\)-action and canonical action form that define a moment map whose zero set is the transverse cotangent bundle \(T_{G}^{*}M\) of \(M\). The authors show that the infinitesimal index in [the authors, ``The infinitesimal index'', Preprint (2010), \url{arXiv:1003.3525}] defines an isomorphism between the equivariant cohomology with compact supports of \(T_{G}^{*}M_{k}\) and a space of splines introduced in [the authors, Transform. Groups 15, No. 4, 751--773 (2010; Zbl 1223.58015)], where similar ideas were expressed in the language of equivariant \(K\)-theory and transversally elliptic operators. The authors plan to use their infinitesimal index to give explicit index formulas for transversally elliptic operators. We propose an epidemic model consisting five compartments within a total population with Crowley-Martin incidence rate and Holling type II treatment, where total population is separated by the susceptible, the vaccinated, the exposed, the infected and the removed in this paper. We firstly prove that the epidemic model admits a unique global positive solution by contradiction. We then find out that diseases tend to extinction provided that the basic reproduction number is less than one. Moreover, the sufficient conditions of persistence for infectious diseases are obtained by constructing suitable Lyapunov functions. | 0 |
We define a graph encoding the structure of contact surgery on contact 3-manifolds and analyse its basic properties and some of its interesting subgraphs. In [Algebr. Geom. Topol. 1, 153--172 (2001; Zbl 0974.53061)] and [Math. Proc. Camb. Philos. Soc. 136, No.~3, 583--598 (2004; Zbl 1069.57015)] the authors introduced the notion of contact Dehn surgery, especially contact \((\pm 1)\)-surgery. In the article under review, a variety of handle moves in the resulting contact surgery diagrams is described: the first and second Kirby move, handle cancellation and handle moves involving 1-handles translated into \((+1)\)-surgeries.
As an application of these handle moves, the authors discuss the classification of loose Legendrian knots. Furthermore they prove a one-to-one correspondence (up to Legendrian isotopy) between long Legendrian knots in 3-space and their completion in the 3-sphere. | 1 |
We define a graph encoding the structure of contact surgery on contact 3-manifolds and analyse its basic properties and some of its interesting subgraphs. This paper sets out a model that simultaneously determines insurers' satisficing compositions of their insurance and investment portfolios. This model can be explained as follows: different insurance lines and investments have different rates of return and different risks associated with those rates of return. Different insurers also have different, but satisfactory levels of return on equity and risk levels of violating the minimum requirement on cash and liquid assets. We propose a chance constrained programming approach to incorporate all of these factors in the portfolio analysis. | 0 |
When a differential graded algebra, \((A, d)\), is strongly homotopy commutative, the first author and \textit{J.-C. Thomas} have proved in [Topology 41, No. 1, 85--106 (2002; Zbl 1011.16008)] that the Hochschild homology \(HH_*A\) is naturally a commutative graded algebra. Here the authors extend this result to the negative cyclic homology \(HC^-_*(A)\) of \((A, d)\). They apply it to the normalized singular cochains \(N^*(X)\) of a space \(X\) and show that the Jones' isomorphism of graded vector spaces is an isomorphism of graded algebras, \(HC^-_* N^*(X)\cong H^{-*}_{S^1}(LX)\). Explicit computations are given when \(X\) is an even sphere or for some particular complex projective spaces. The authors construct a product on the normalized Hochschild complex of a graded algebra if it is strongly homotopy commutative (rather than the usual condition of being strictly commutative). The application they have in mind is the algebra of normalized singular cochains on a simply connected space \(X\) with coefficients in a field \(k\), \(N^*(X;k)\), which admits a natural strongly homotopy commutative algebra structure by work of \textit{H. J. Munkholm} [J. Pure Appl. Algebra 5, 1-50 (1974; Zbl 0294.55011)]. \textit{J. D. S. Jones} has shown [Invent. Math. 87, 403-423 (1987; Zbl 0644.55005)] that \(HH_*(N^*(X;k))\cong H^*(X^{S^1};k)\) as graded vector spaces, and the authors show that their product on the left term agrees with the usual cup product on the right. Using a notion of equivalence of strongly homotopy commutative algebras, they are able to replace \(N^*(X;k)\) with commutative DGAs in various situations to facilitate calculations. In particular they give results on the multiplicative structure of \(H^*(X^{S^1};\mathbb{F}_p)\) for \(X=S^{2n}\), \(\mathbb{C} P^n\), and also \(X=\Sigma\mathbb{C} P^n\) for \(p=2\) and \(X=G_2\) for \(p=5\). | 1 |
When a differential graded algebra, \((A, d)\), is strongly homotopy commutative, the first author and \textit{J.-C. Thomas} have proved in [Topology 41, No. 1, 85--106 (2002; Zbl 1011.16008)] that the Hochschild homology \(HH_*A\) is naturally a commutative graded algebra. Here the authors extend this result to the negative cyclic homology \(HC^-_*(A)\) of \((A, d)\). They apply it to the normalized singular cochains \(N^*(X)\) of a space \(X\) and show that the Jones' isomorphism of graded vector spaces is an isomorphism of graded algebras, \(HC^-_* N^*(X)\cong H^{-*}_{S^1}(LX)\). Explicit computations are given when \(X\) is an even sphere or for some particular complex projective spaces. In this paper, we propose and analyze a distributed negotiation strategy for a multi-agent, multi-attribute negotiation in which the agents have no information about the utility functions of other agents. We analytically prove that, if the zone of agreement is nonempty and the agents concede up to their reservation utilities, agents generating offers using our offer-generation strategy, namely the sequential projection strategy, will converge to an agreement acceptable to all the agents; the convergence property does not depend on the specific concession strategy. In considering agents' incentive to concede during the negotiation, we propose and analyze a reactive concession strategy. Through computational experiments, we demonstrate that our distributed negotiation strategy yields performance sufficiently close to the Nash bargaining solution and that our algorithms are robust to potential deviation strategies. Methodologically, our paper advances the state of the art of alternating projection algorithms, in that we establish the convergence for the case of multiple, moving sets (as opposed to two static sets in the current literature). Our paper introduces a new analytical foundation for a broad class of computational group decision and negotiation problems. | 0 |
The authors study stochastic estimation of the trace of a matrix function \(f(A)\), where \(A\) and \(f(A)\in \mathbb{C}\), and \(f:z\in D\subseteq\mathbb{C}\to f(z)\in\mathbb{C}\) is an operator. Main focus is on the inverse operator \(f(z)=z^{-1}\).
Unfortunately, it is usually unfeasible to calculate diagonal elements of \(f(A)\) directly. One of the possibilities how to overcome this problem is to use so called deterministic approximation techniques. Instead, in this paper the authors concentrate on the use of stochastic Monte Carlo approximations. The approach used can be regarded as a variance reduction technique applied to the Hutchinson estimator [\textit{M. F. Hutchinson}, Commun. Stat., Simulation Comput. 19, No. 2, 433--450 (1990; Zbl 0718.62058)]
\[
tr(f(A)) \approx \sum_1^N (x^{(n)})^\ast f(A)x^{(n)},
\]
where the components of the random vectors \(x^{(n)}\) obey an appropriate probability distribution. Unfortunately, the variance of the estimator decreases only like \(O(N^{-1})\), which makes the method too costly when higher precision are to be achieved. Thus, the multilevel approach aims at curing this by working with representations of \(A\) at different levels. On the higher numbered levels, evaluating \(f(A)\) becomes increasingly cheap, while on the lower levels, which are more costly to evaluate, the variance is small. The focus is on the trace of the matrix inverse, where one can evaluate \(A^{-1}x\) using a fast solver. Note that for a general matrix function \(f(A)\), stochastic trace estimation techniques can be combined with the Lanczos process to approximately evaluate the quadratic forms \(x^\ast f(A)x\).
The paper is organized as follows: In Section 2, general framework of multilevel Monte Carlo estimators is recalled. In Section 3, Hutchinson's method for stochastically estimating the trace is discussed. New multilevel approach and comparison to known approaches is covered in Section 4. Finally, several numerical results are presented in Section 5. An unbiased stochastic estimator of tr(I-A), where A is the influence matrix associated with the calculation of Laplacian smoothing splines, is described. The estimator satisfies a minimum variance criterion and does not require the simulation of a standard normal variable. It uses instead simulations of the discrete random variable which takes the vaues 1, -1 each with probability 1/2. Bounds on the variance of the estimator are obtained using elementary methods. The estimator can be used to approximately minimize generalized cross validation (GCV) when using discretized iterative methods for fitting Laplacian smoothing splines to very large data sets. | 1 |
The authors study stochastic estimation of the trace of a matrix function \(f(A)\), where \(A\) and \(f(A)\in \mathbb{C}\), and \(f:z\in D\subseteq\mathbb{C}\to f(z)\in\mathbb{C}\) is an operator. Main focus is on the inverse operator \(f(z)=z^{-1}\).
Unfortunately, it is usually unfeasible to calculate diagonal elements of \(f(A)\) directly. One of the possibilities how to overcome this problem is to use so called deterministic approximation techniques. Instead, in this paper the authors concentrate on the use of stochastic Monte Carlo approximations. The approach used can be regarded as a variance reduction technique applied to the Hutchinson estimator [\textit{M. F. Hutchinson}, Commun. Stat., Simulation Comput. 19, No. 2, 433--450 (1990; Zbl 0718.62058)]
\[
tr(f(A)) \approx \sum_1^N (x^{(n)})^\ast f(A)x^{(n)},
\]
where the components of the random vectors \(x^{(n)}\) obey an appropriate probability distribution. Unfortunately, the variance of the estimator decreases only like \(O(N^{-1})\), which makes the method too costly when higher precision are to be achieved. Thus, the multilevel approach aims at curing this by working with representations of \(A\) at different levels. On the higher numbered levels, evaluating \(f(A)\) becomes increasingly cheap, while on the lower levels, which are more costly to evaluate, the variance is small. The focus is on the trace of the matrix inverse, where one can evaluate \(A^{-1}x\) using a fast solver. Note that for a general matrix function \(f(A)\), stochastic trace estimation techniques can be combined with the Lanczos process to approximately evaluate the quadratic forms \(x^\ast f(A)x\).
The paper is organized as follows: In Section 2, general framework of multilevel Monte Carlo estimators is recalled. In Section 3, Hutchinson's method for stochastically estimating the trace is discussed. New multilevel approach and comparison to known approaches is covered in Section 4. Finally, several numerical results are presented in Section 5. In this paper, we extend the single-step pseudo-spectral method for nonlinear Volterra integral equations of the second kind to the multistep pseudo-spectral method. We also analyze the convergence of the \(hp\)-version of the multistep pseudo-spectral method under the \(L^2\)-norm, and the result shows that the scheme enjoys high order accuracy and can be implemented in a stable and efficient manner. In addition, it is very suitable for long time calculations and large step size situations. Numerical experiments confirm the theoretical expectations. | 0 |
The main result is that any state on an effect algebra can be represented as an integral through a regular Borel probability measure. This generalizes the results obtained for Łukasiewicz tribes in [\textit{D. Butnariu} and \textit{E. P. Klement}, Triangular Norm Based Measures and Games with Fuzzy Coalitions.Theory and Decision Library. Series C: Game Theory, Mathematical Programming and Operations Research. 10. Dordrecht: Kluwer Academic Publishers. (1993; Zbl 0804.90145)] and for interval effect algebras in [\textit{A. Dvurečenskij}, Every state on interval effect algebra is integral, J. Math. Phys. 51, 083508-12 (2010)]. Further, the convex structure of the state space is studied. It is shown that every \(\sigma\)-convex combination of extremal states on a monotone \(\sigma\)-complete effect algebra is a Jauch--Piron state. The book presents in a compact and consistent form a relatively wide scale of topics. The briefly described concepts of triangular norm and conorm are used to introduce the fundamental notion of \(t\)-norm-based measure and to describe its properties. Such measures are used to define and study games with fuzzy coalitions and their solutions where special attention is paid to the Aumann-Shapley value and its diagonal modification. The fuzzy coalitions are defined as fuzzy subsets of a general non-empty set of players where the values of the membership functions represent the share of individual participation of players in particular coalitions. The last chapter of the book deals with a few applications of the general theory (games with crisp coalitions, economic equilibria, plausibility measures, etc.). The book is completed by an index and representative bibliography. | 1 |
The main result is that any state on an effect algebra can be represented as an integral through a regular Borel probability measure. This generalizes the results obtained for Łukasiewicz tribes in [\textit{D. Butnariu} and \textit{E. P. Klement}, Triangular Norm Based Measures and Games with Fuzzy Coalitions.Theory and Decision Library. Series C: Game Theory, Mathematical Programming and Operations Research. 10. Dordrecht: Kluwer Academic Publishers. (1993; Zbl 0804.90145)] and for interval effect algebras in [\textit{A. Dvurečenskij}, Every state on interval effect algebra is integral, J. Math. Phys. 51, 083508-12 (2010)]. Further, the convex structure of the state space is studied. It is shown that every \(\sigma\)-convex combination of extremal states on a monotone \(\sigma\)-complete effect algebra is a Jauch--Piron state. The author states that this paper considers a special case of the following problem: Suppose that \(\kappa\) is an infinite cardinal. Characterize those \(X\subseteq \kappa^+\) such that \(X\) contains a closed unbounded (club) subset of \(\kappa^+\) in some \(\kappa\) and \(\kappa^+\) preserving outer model. (\(W\) is called an outer model of \(V\) if \(W\) and \(V\) are transitive standard models of ZFC and \(V\cap OR= W\cap OR\).) The principal result is the following Theorem: Assume \(\text{GCH}+\square\). There is a definable operation from bounded pattern width subsets \(X\) of \(\aleph_{\omega+ 1}\) to trees \(T_X\) such that the following three statements are equivalent.
(1) There is a club subset of \(\aleph_{\omega+ 1}\) contained in \(X\) in some covered \(\aleph_\omega\) and \(\aleph_{\omega+ 1}\) preserving outer model,
(2) \(T_X\) is ill-founded,
(3) There is a club subset of \(\aleph_{\omega+ 1}\) contained in \(X\) in a fully covered, GCH, \(\aleph_\omega\) and \(\aleph_{\omega+ 1}\) preserving set forcing extension.
The definitions of the new notions mentioned here are too complex to be explained in this review. | 0 |
Let \((X,P)\) be a metric space and \(A\), \(X_0\) be subsets of \(X\). \(A\) is said to be Čebyšev in \(X\) (Čebyšev relative to \(X_0\)) if for every \(x\in X\) \((x\in X_0)\) there is a unique nearest point in \(A\). A closed convex subset of the Euclidean space \(\mathbb{R}^n\) is said to be strictly convex if its boundary does not contain any segment. Let \(C^n\) denote the class of non-empty compact subsets of \(\mathbb{R}^n\) and \(K^n\) the class of non-empty compact convex subsets of \(\mathbb{R}^n\). The authors study Čebyšev sets in some hyperspaces over \(\mathbb{R}^n\), endowed with the Hausdorff metric, mainly the hyperspaces \(C^n\) of compact sets, \(K^n\) of compact convex sets, and of strictly convex compact sets. They also disprove two conjectures (6.2 and 6.4) from \textit{A. Bogdewicz} and \textit{M. Moszyńska} [Čebyšev sets in the space of convex bodies. Supplemento ai Rendiconti del Circolo Matemático di Palermo. Serie II 77, 19--39 (2006; Zbl 1114.52002)] concerning relationships between convexity and Čebyšev in \(K^n\). Some new conjectures have also been made in the paper. Authors' abstract: A set \(A\) in a metric space is a Čebyšev set if every point has a unique nearest point in \(A\). In \(R^n\) the class of Čebyšev sets coincides with the class of nonempty closed convex sets.
The main question, we try to answer in the present paper, is the following: is there an analogue of this coincidence in the space \({\mathcal K}^n\) of all compact convex subsets of \(R^n\) or in the space \({\mathcal K}_0^n\) of convex bodies, with the Hausdorff metric? We first consider Čebyšev sets in \({\mathcal K}_0^n\) and \({\mathcal K}^n\) invariant under similarities (Sections 2 and 3). In Sections 4 and 5 we show that affine convex subsets of \({\mathcal K}^n\) need not be Čebyšev sets, while strict affine convexity is sufficient. | 1 |
Let \((X,P)\) be a metric space and \(A\), \(X_0\) be subsets of \(X\). \(A\) is said to be Čebyšev in \(X\) (Čebyšev relative to \(X_0\)) if for every \(x\in X\) \((x\in X_0)\) there is a unique nearest point in \(A\). A closed convex subset of the Euclidean space \(\mathbb{R}^n\) is said to be strictly convex if its boundary does not contain any segment. Let \(C^n\) denote the class of non-empty compact subsets of \(\mathbb{R}^n\) and \(K^n\) the class of non-empty compact convex subsets of \(\mathbb{R}^n\). The authors study Čebyšev sets in some hyperspaces over \(\mathbb{R}^n\), endowed with the Hausdorff metric, mainly the hyperspaces \(C^n\) of compact sets, \(K^n\) of compact convex sets, and of strictly convex compact sets. They also disprove two conjectures (6.2 and 6.4) from \textit{A. Bogdewicz} and \textit{M. Moszyńska} [Čebyšev sets in the space of convex bodies. Supplemento ai Rendiconti del Circolo Matemático di Palermo. Serie II 77, 19--39 (2006; Zbl 1114.52002)] concerning relationships between convexity and Čebyšev in \(K^n\). Some new conjectures have also been made in the paper. In this paper, we define a class of analytic functions by using the concept of complex order. This class of analytic functions generalizes the class of Bazilevič functions. In the present work, we derive various useful properties and characteristics of this class such as coefficient bounds, Fekete-Szegö type inequality, arclength, integral preserving property, radius problem and some other interesting properties. Relevant connections of the results presented here with those obtained in earlier works are pointed. | 0 |
In a Banach space \(X\), an element \(x\) is said to be orthogonal to a subspace \(M\) if \(\|x\|\) is equal to \(\text{dist}(x,M)\). Hence a finite set being orthogonal means that none of the elements can be very close to, i.e., be almost linearly dependent of, the subspace spanned by the other elements. A condition number measures `how orthogonal' the system is. Such a condition number is first defined on a finite-dimensional Euclidean space and then generalized to a concept of uniform conditioning of an infinite sequence. Here these ideas are used in the context of solving integral equations. In particular, it involves bases of spectral subspaces of a bounded operator \(F\) on \(X\). In practice, \(F\) is approximated numerically by a sequence of operators \(T_n=S_n+U_n\). The sequence \(S_n\) converges to a finite rank operator and \(U_n\) to some operator satisfying an invariance condition [\textit{B. V. Limaye}, Oper. Matrices 6, No.~2, 357--370 (2012; Zbl 1269.47013)]. A well conditioned basis of the spectral subspace of \(T_n\) is constructed, first in the approximating Euclidean space \(\mathbb{C}^n\) and then an estimate is given for the deterioration of the conditioning when the finite-dimensional system is recast into the setting of \(X\). This is analysed for specific approximation and lifting operators \(K\) and \(L\) in the process \(X\overset{K}{\longrightarrow}\mathbb{C}^n\overset{L}{\longrightarrow}X\). All this and more leads to the construction of a uniformly conditioned sequence for this problem, in particular, for the case where \(F\) is an integral operator on \(C([0,1])\) with weakly singular kernel or for operators on \(L^\infty([a,b])\) that are the sum of integral operators with continuous kernel and multiplication by a continuous function. Let \(X\) be a Banach space of dimension \(d\leq \infty\). Then a linear operator \(S\) on \(X\) of finite rank \(n\) can be written as \(Sx=\sum_{j=1}^n f_j(x)x_j\) with \(x_j\in X\) and \(f_j\) linear functionals on \(X\). So we can think of it as a matrix \(S\in \mathbb{C}^{d\times d}\) that can be written as \(S=LK\) with analysis operator \(K\in\mathbb{C}^{n\times d}\) whose rows represent the \(f_j\) and synthesis operator \(L:\mathbb{C}^{d\times n}\to X\) whose columns represent the \(x_j\). This operator \(S\) on \(X\) can be associated with a matrix \(A\in\mathbb{C}^{n\times n}\) by \(A=KL\), hence satisfying \(AK=KS\). Let \(U\) and \(V\) be other operators on \(X\) of infinite rank that satisfy some invariance properties, namely \(CK=KU\) and \(LD=VL\) for some matrices \(C,D\in\mathbb{C}^{n\times n}\). This paper relates the spectral properties of the operator \(T=S+U\) (or \(T=S+V\)) on \(X\) and its \(n\)-dimensional counterpart, the matrix \(B=A+C\) (or \(B=A+D\)). This allows to approximate a large \(d\)-dimensional problem for a low rank \(n\) operator (\(S\)) with particular high rank perturbations (\(U\) or \(V\)) by a low \(n\)-dimensional matrix problem (\(B\)). | 1 |
In a Banach space \(X\), an element \(x\) is said to be orthogonal to a subspace \(M\) if \(\|x\|\) is equal to \(\text{dist}(x,M)\). Hence a finite set being orthogonal means that none of the elements can be very close to, i.e., be almost linearly dependent of, the subspace spanned by the other elements. A condition number measures `how orthogonal' the system is. Such a condition number is first defined on a finite-dimensional Euclidean space and then generalized to a concept of uniform conditioning of an infinite sequence. Here these ideas are used in the context of solving integral equations. In particular, it involves bases of spectral subspaces of a bounded operator \(F\) on \(X\). In practice, \(F\) is approximated numerically by a sequence of operators \(T_n=S_n+U_n\). The sequence \(S_n\) converges to a finite rank operator and \(U_n\) to some operator satisfying an invariance condition [\textit{B. V. Limaye}, Oper. Matrices 6, No.~2, 357--370 (2012; Zbl 1269.47013)]. A well conditioned basis of the spectral subspace of \(T_n\) is constructed, first in the approximating Euclidean space \(\mathbb{C}^n\) and then an estimate is given for the deterioration of the conditioning when the finite-dimensional system is recast into the setting of \(X\). This is analysed for specific approximation and lifting operators \(K\) and \(L\) in the process \(X\overset{K}{\longrightarrow}\mathbb{C}^n\overset{L}{\longrightarrow}X\). All this and more leads to the construction of a uniformly conditioned sequence for this problem, in particular, for the case where \(F\) is an integral operator on \(C([0,1])\) with weakly singular kernel or for operators on \(L^\infty([a,b])\) that are the sum of integral operators with continuous kernel and multiplication by a continuous function. We use an event study approach to estimate the burden of the financial regulations associated with Systemically Important Financial Institution (SIFI) designation. On March 30, 2016, the US District Court determined that MetLife's SIFI designation was arbitrary and capricious because the Financial Stability Oversight Council (FSOC) failed to weigh the economic cost of the financial regulation on MetLife against the benefits of increased financial stability. We find significant positive abnormal returns for MetLife and AIG on the date of the ruling. We estimate that the lifting of the SIFI designation created 1.4 billion in corporate wealth for MetLife, suggesting that MetLife would be 3.4\% more profitable as a non-SIFI. These gains fall short of the 8 billion stipulated by MetLife in its complaint. We also find significant abnormal returns to SIFI institutions on the day following the US Presidential election. | 0 |
A mapping \(f\) of an open subset of \(\mathbb{R}^ n\) into \(\mathbb{R}^ n\) is called a \(p\)-harmonic morphism if \(u \circ f\) is \(p\)-harmonic in the preimage \(f^{-1}(d)\) whenever \(u\) is \(p\)-harmonic in \(D\). Recall that \(u\) is \(p\)-harmonic in an open set \(\Omega\) if \(u \in C(\Omega) \cap W^{1,p}(\Omega)\) satisfies the \(p\)-Laplace equation
\[
\text{div}(| \nabla u|^{p - 2} \nabla u)=0
\]
in \(\Omega\); here \(1 < p < \infty\). The case \(p = 2\) is the classical Laplacian; then harmonic morphisms are known to be analytic or antianalytic mappings in the planar case, and similarity transformations (i.e. compositions of rotations and translations) in higher dimensions, and vice versa.
For a general \(p\), it is easily seen that the \(p\)-Laplacian is invariant under similarity transformations and, moreover, the \(n\)-Laplacian is invariant under all 1-quasiregular mappings (i.e. Möbius maps if \(n \geq 3\)). Therefore these mappings are \(p\)-harmonic morphisms.
In the present paper the authors show that a discrete \(p\)-harmonic morphism necessarily is 1-quasiregular if \(p = n\) and a similarity if \(p \neq n\). The paper complements the paper [\textit{J. Heinonen, T. Kilpelaeinen} and \textit{O. Martio}, Nagoya Math. J. 125, 115-140 (1992; Zbl 0776.31007)], where harmonic morphisms between nonlinear harmonic spaces are characterized. For fixed \(1<p< \infty\), let \(A_ p\) denote the family of all mappings \(a:\mathbb{R}^ n \times \mathbb{R}^ n \to \mathbb{R}^ n\), \((x,h) \to a(x,h)\), satisfying appropriate conditions (such as being comparable to \(| h |^{p-1})\). A function \(u\) defined in an open set \(\Omega \subset \mathbb{R}^ n\) is said to be \(a\)-harmonic in \(\Omega\) if it is a continuous (generalized) solution in \(\Omega\) of the quasilinear elliptic equation \(-\text{div} a(x,\nabla u)=0\).
Definition: Let \(a^*\) and \(a\) belong to \(A_ p\). A continuous mapping \(f:\Omega \to \mathbb{R}^ n\) is an \((a^*,a)\)-harmonic morphism if \(u \circ f\) is \(a^*\)-harmonic in \(f^{-1}(\Omega)\) whenever \(u\) is \(a\)-harmonic in \(\Omega\). \(f\) is said to be an \(A_ p\)-harmonic morphism if \(f\) is an \((a^*,a)\)-harmonic morphism for some \(a^*\) and \(a\) in \(A_ p\).
Definition: A continuous mapping \(f:\Omega \to \mathbb{R}^ n\) is \(K\)-quasi- regular if it belongs to the Sobolev space \(W^{1,n}_{\text{loc}}(\Omega)\) and satisfies the inequality \(| f'(x)|^ n\leftrightharpoons KJ_ f(x)\) a.e. in \(\Omega\), where \(f'(x)\) is the formal derivative matrix and \(J_ f(x)\) is the Jacobian determinant.
This paper is an extensive study of \(A_ p\)-harmonic morphisms especially in the cae \(p=n\) where the authors prove that every sense- preserving \(A_ n\)-harmonic morphism is quasi-regular. Also, if \(1<p<n\) and \(f:\Omega \to \mathbb{R}^ n\) is a (suitably restricted) \(A_ p\)- harmonic morphism, then \(f\) is of bounded length distortion in every compact set of \(\Omega\). There is a section concerning asymptotic estimates of \(a\)-harmonic functions near an essential isolated singularity. There are quite a few references in the text to closely related work upon which the present proofs depend. | 1 |
A mapping \(f\) of an open subset of \(\mathbb{R}^ n\) into \(\mathbb{R}^ n\) is called a \(p\)-harmonic morphism if \(u \circ f\) is \(p\)-harmonic in the preimage \(f^{-1}(d)\) whenever \(u\) is \(p\)-harmonic in \(D\). Recall that \(u\) is \(p\)-harmonic in an open set \(\Omega\) if \(u \in C(\Omega) \cap W^{1,p}(\Omega)\) satisfies the \(p\)-Laplace equation
\[
\text{div}(| \nabla u|^{p - 2} \nabla u)=0
\]
in \(\Omega\); here \(1 < p < \infty\). The case \(p = 2\) is the classical Laplacian; then harmonic morphisms are known to be analytic or antianalytic mappings in the planar case, and similarity transformations (i.e. compositions of rotations and translations) in higher dimensions, and vice versa.
For a general \(p\), it is easily seen that the \(p\)-Laplacian is invariant under similarity transformations and, moreover, the \(n\)-Laplacian is invariant under all 1-quasiregular mappings (i.e. Möbius maps if \(n \geq 3\)). Therefore these mappings are \(p\)-harmonic morphisms.
In the present paper the authors show that a discrete \(p\)-harmonic morphism necessarily is 1-quasiregular if \(p = n\) and a similarity if \(p \neq n\). The paper complements the paper [\textit{J. Heinonen, T. Kilpelaeinen} and \textit{O. Martio}, Nagoya Math. J. 125, 115-140 (1992; Zbl 0776.31007)], where harmonic morphisms between nonlinear harmonic spaces are characterized. A novel version of spectral mapping for partially labeled sample classification is proposed in this paper. This new method adds the label information into the mapping process, and adopts the geodesic distance rather than Euclidean distance as the measure of the difference between two data points. The experimental results show that the proposed method yields significant benefits for partially labeled classification with respect to the previous methods. | 0 |
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