text: 32 A chordal graph G K \ G’s clique tree Intersection graph of C { } K Fig. 14: Let C be the clique f, g, h, i and let be the remaining cliques of G. K To the lower right is the intersection graph of the members of after elements of C have been removed from them. This graph is connected, which means by { } Lemma 23 that f, g, h, i cannot lie on the path between any pair of members K of in any clique tree of G. T Lemma 24. Let be a clique tree for a chordal graph G and let K be a leaf. Then K contains a simplicial vertex of G. Let S be the simplicial vertices of K T (cid:48) T and let be the result of deleting leaf K from . Deleting S from G yields an T (cid:48) induced subgraph that has as a clique tree. (See Figure 15.) T A chordal graph G G’s clique tree − { } − { } G h, i G h, i ’s clique tree T Fig. 15: “Shrinking” a clique tree. If G is a chordal graph and is a clique tree { } { } T for it, then deleting the simplicial vertices h, i in a leaf f, g, h, i of gives a T smaller graph that has as a clique tree the tree obtained by deleting K from . ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: A

text: chordal

text: graph

text: G

text: K

text: G’s

text: clique

text: tree

text: Intersection

text: graph

text: of

text: C

text: K

text: Fig.

text: Let

text: C

text: be

text: the

text: clique

text: f,

text: g,

text: h,

text: i

text: and

text: let

text: be

text: the

text: remaining

text: cliques

text: of

text: G.

text: K

text: To

text: the

text: lower

text: right

text: is

text: the

text: intersection

text: graph

text: of

text: the

text: members

text: of

text: after

text: elements

text: of

text: C

text: have

text: been

text: removed

text: from

text: them.

text: This

text: graph

text: is

text: connected,

text: which

text: means

text: by

text: Lemma

text: that

text: f,

text: g,

text: h,

text: i

text: cannot

text: lie

text: on

text: the

text: path

text: between

text: any

text: pair

text: of

text: members

text: K

text: of

text: in

text: any

text: clique

text: tree

text: of

text: G.

text: T

text: Lemma

text: Let

text: be

text: a

text: clique

text: tree

text: for

text: a

text: chordal

text: graph

text: G

text: and

text: let

text: K

text: be

text: a

text: leaf.

text: Then

text: K

text: contains

text: a

text: simplicial

text: vertex

text: of

text: G.

text: Let

text: S

text: be

text: the

text: simplicial

text: vertices

text: of

text: K

text: T

text: T

text: and

text: let

text: be

text: the

text: result

text: of

text: deleting

text: leaf

text: K

text: from

text: Deleting

text: S

text: from

text: G

text: yields

text: an

text: T

text: induced

text: subgraph

text: that

text: has

text: as

text: a

text: clique

text: tree.

text: (See

text: Figure

text: T

text: A

text: chordal

text: graph

text: G

text: G’s

text: clique

text: tree

text: G

text: h,

text: i

text: G

text: h,

text: i

text: ’s

text: clique

text: tree

text: T

text: Fig.

text: “Shrinking”

text: a

text: clique

text: tree.

text: If

text: G

text: is

text: a

text: chordal

text: graph

text: and

text: is

text: a

text: clique

text: tree

text: T

text: for

text: it,

text: then

text: deleting

text: the

text: simplicial

text: vertices

text: h,

text: i

text: in

text: a

text: leaf

text: f,

text: g,

text: h,

text: i

text: of

text: gives

text: a

text: T

text: smaller

text: graph

text: that

text: has

text: as

text: a

text: clique

text: tree

text: the

text: tree

text: obtained

text: by

text: deleting

text: K

text: from

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 6 The paper also makes extensive use of generic techniques whose usefulness, to our knowledge, has not been previously recognized. An example is the extensive use of Lemma 1 in various parts of the algorithm. Tucker recognized a relationship between the LB graphs and the Tucker matrices: a Tucker submatrix occurs in the clique matrix of every LB graph. The relationship between Tucker submatrices of clique matrices of graphs and their induced LB subgraphs, however, has a much richer structure than has been previously recognized, which is shown in the last section of the paper. An open question is whether a minimal, unique, especially simple, or oth- erwise interesting special case of the certiﬁcate from [12] can be obtained by applying the algorithm of that paper to a Tucker submatrix obtained by the al- gorithm of the present paper. Tucker submatrices may be useful in heuristics for ﬁnding large submatrices that have the consecutive-ones property, small Tucker matrices, or identifying errors in biological data [21, 3]. Our techniques provide new tools for such heuristics. The results of this paper appeared in preliminary form in [17]. 2 Preliminaries Given a graph G, let V denote the set of vertices and E denote the set of | | | | ∅ ⊂ ⊆ edges. Let n denote V and m denote E . If X V , let G[X] denote the − \ { } − subgraph induced by X. By G x, we denote G[V x ], and by G X, we \ denote G[V X]. Given an ordered sequence (a , a , . . . , a ), let a preﬁx of the sequence denote 1 2 k ≤ ≤ a consecutive subsequence (a , a , . . . , a ) for some i such that 0 i k. (If 1 2 i i = 0, the preﬁx is empty.) Similarly, let a suﬃx denote (a , a . . . a ) for some ≤ ≤ i i+1 k i such that 1 i k + 1. We treat the rows as sets, where each row R is the set of columns in which the row has a 1. We will use calligraphic font for collections of sets, hence for sets of rows. R R For a set of rows and a set C of columns of a matrix, let [C] denote the { ∩ | ∈ R} R R restriction R C R of to columns in C. Suppose is the set of rows of a consecutive-ones ordered matrix and (c , c , . . . , c ) is the ordering of the 1 2 m columns. In linear time, we can ﬁnd, for each row, the leftmost and rightmost column in the row. Let us call these the left endpoint and right endpoint of the row. Every graph G is the intersection graph of rows of the clique matrix, since two vertices of G are adjacent if and only if they are members of a common clique. Therefore, a consecutive-ones ordering of the clique matrix of a graph gives an interval model of the graph; the intervals are the consecutive blocks of 1’s in the rows. That interval graphs are a subclass of the class of chordal graphs follows from inclusion of the G ’s among the LB subgraphs. III When a graph is chordal, the problem of deciding whether it is an interval graph reduces to the problem of deciding whether its clique matrix has the consecutive-ones property. When G is chordal, the algorithm of Rose, Tarjan

label: paragraph

text: The

text: paper

text: also

text: makes

text: extensive

text: use

text: of

text: generic

text: techniques

text: whose

text: usefulness,

text: to

text: our

text: knowledge,

text: has

text: not

text: been

text: previously

text: recognized.

text: An

text: example

text: is

text: the

text: extensive

text: use

text: of

text: Lemma

text: in

text: various

text: parts

text: of

text: the

text: algorithm.

text: Tucker

text: recognized

text: a

text: relationship

text: between

text: the

text: LB

text: graphs

text: and

text: the

text: Tucker

text: matrices:

text: a

text: Tucker

text: submatrix

text: occurs

text: in

text: the

text: clique

text: matrix

text: of

text: every

text: LB

text: graph.

text: The

text: relationship

text: between

text: Tucker

text: submatrices

text: of

text: clique

text: matrices

text: of

text: graphs

text: and

text: their

text: induced

text: LB

text: subgraphs,

text: however,

text: has

text: a

text: much

text: richer

text: structure

text: than

text: has

text: been

text: previously

text: recognized,

text: which

text: is

text: shown

text: in

text: the

text: last

text: section

text: of

text: the

text: paper.

text: An

text: open

text: question

text: is

text: whether

text: a

text: minimal,

text: unique,

text: especially

text: simple,

text: or

text: oth-

text: erwise

text: interesting

text: special

text: case

text: of

text: the

text: certiﬁcate

text: from

text: can

text: be

text: obtained

text: by

text: applying

text: the

text: algorithm

text: of

text: that

text: paper

text: to

text: a

text: Tucker

text: submatrix

text: obtained

text: by

text: the

text: al-

text: gorithm

text: of

text: the

text: present

text: paper.

text: Tucker

text: submatrices

text: may

text: be

text: useful

text: in

text: heuristics

text: for

text: ﬁnding

text: large

text: submatrices

text: that

text: have

text: the

text: consecutive-ones

text: property,

text: small

text: Tucker

text: matrices,

text: or

text: identifying

text: errors

text: in

text: biological

text: data

text: Our

text: techniques

text: provide

text: new

text: tools

text: for

text: such

text: heuristics.

text: The

text: results

text: of

text: this

text: paper

text: appeared

text: in

text: preliminary

text: form

text: in

text: Preliminaries

text: Given

text: a

text: graph

text: G,

text: let

text: V

text: denote

text: the

text: set

text: of

text: vertices

text: and

text: E

text: denote

text: the

text: set

text: of

text: edges.

text: Let

text: n

text: denote

text: V

text: and

text: m

text: denote

text: E

text: If

text: X

text: V

text: let

text: G[X]

text: denote

text: the

text: subgraph

text: induced

text: by

text: X.

text: By

text: G

text: x,

text: we

text: denote

text: G[V

text: x

text: and

text: by

text: G

text: X,

text: we

text: denote

text: G[V

text: X].

text: Given

text: an

text: ordered

text: sequence

text: (a

text: a

text: a

text: let

text: a

text: preﬁx

text: of

text: the

text: sequence

text: denote

text: k

text: a

text: consecutive

text: subsequence

text: (a

text: a

text: a

text: for

text: some

text: i

text: such

text: that

text: i

text: k.

text: (If

text: i

text: i

text: the

text: preﬁx

text: is

text: empty.)

text: Similarly,

text: let

text: a

text: suﬃx

text: denote

text: (a

text: a

text: a

text: for

text: some

text: i

text: k

text: i

text: such

text: that

text: i

text: k

text: We

text: treat

text: the

text: rows

text: as

text: sets,

text: where

text: each

text: row

text: R

text: is

text: the

text: set

text: of

text: columns

text: in

text: which

text: the

text: row

text: has

text: a

text: We

text: will

text: use

text: calligraphic

text: font

text: for

text: collections

text: of

text: sets,

text: hence

text: for

text: sets

text: of

text: rows.

text: R

text: R

text: For

text: a

text: set

text: of

text: rows

text: and

text: a

text: set

text: C

text: of

text: columns

text: of

text: a

text: matrix,

text: let

text: [C]

text: denote

text: the

text: R}

text: R

text: R

text: restriction

text: R

text: C

text: R

text: of

text: to

text: columns

text: in

text: C.

text: Suppose

text: is

text: the

text: set

text: of

text: rows

text: of

text: a

text: consecutive-ones

text: ordered

text: matrix

text: and

text: (c

text: c

text: c

text: is

text: the

text: ordering

text: of

text: the

text: m

text: columns.

text: In

text: linear

text: time,

text: we

text: can

text: ﬁnd,

text: for

text: each

text: row,

text: the

text: leftmost

text: and

text: rightmost

text: column

text: in

text: the

text: row.

text: Let

text: us

text: call

text: these

text: the

text: left

text: endpoint

text: and

text: right

text: endpoint

text: of

text: the

text: row.

text: Every

text: graph

text: G

text: is

text: the

text: intersection

text: graph

text: of

text: rows

text: of

text: the

text: clique

text: matrix,

text: since

text: two

text: vertices

text: of

text: G

text: are

text: adjacent

text: if

text: and

text: only

text: if

text: they

text: are

text: members

text: of

text: a

text: common

text: clique.

text: Therefore,

text: a

text: consecutive-ones

text: ordering

text: of

text: the

text: clique

text: matrix

text: of

text: a

text: graph

text: gives

text: an

text: interval

text: model

text: of

text: the

text: graph;

text: the

text: intervals

text: are

text: the

text: consecutive

text: blocks

text: of

text: in

text: the

text: rows.

text: That

text: interval

text: graphs

text: are

text: a

text: subclass

text: of

text: the

text: class

text: of

text: chordal

text: graphs

text: follows

text: from

text: inclusion

text: of

text: the

text: G

text: ’s

text: among

text: the

text: LB

text: subgraphs.

text: III

text: When

text: a

text: graph

text: is

text: chordal,

text: the

text: problem

text: of

text: deciding

text: whether

text: it

text: is

text: an

text: interval

text: graph

text: reduces

text: to

text: the

text: problem

text: of

text: deciding

text: whether

text: its

text: clique

text: matrix

text: has

text: the

text: consecutive-ones

text: property.

text: When

text: G

text: is

text: chordal,

text: the

text: algorithm

text: of

text: Rose,

text: Tarjan


text: 8 R Fig. 5: A set of rows of a consecutive-ones matrix whose overlap graph Q { } is connected, and its Venn classes, V , V , V , . . . , V . Each element of { } U 1 2 7 V , V , . . . , V is consecutive, their union is consecutive, and the order of V 1 2 7 R 1 through V is uniquely constrained in any consecutive-ones ordering of , up to 7 Q reversal. These are the constrained classes. Columns in the unconstrained class, V , can go on either end of this sequence. U a suﬃx is a subpath that contains its last row. On the other hand, we can treat P ∪ { } it as an unordered collection of sets, as in the expression Z , where Z is a P row not in . 3 Breadth-ﬁrst search on the overlap graph of the rows of a matrix, given a consecutive-ones ordering One step of our algorithm is to ﬁnd a shortest path of the overlap graph of the rows of a consecutive-ones ordered matrix. The diﬃculty in performing BFS on the overlap graph within the O(size(M)) time bound is that the size of the overlap graph is not O(size(M)). A simple example of this is when M has three columns, n/2 rows with 1’s in the ﬁrst and second column and n/2 rows with 1’s in the second and third column. Then size(M) = O(n) but its overlap graph is a complete bipartite graph with n2/4 = Ω(n2) edges. The algorithm is given as Algorithm 1. Lemma 3. The parent relation assigned by OverlapBFS(M, S) (Algorithm 1) is a BFS tree, rooted at S, in the overlap graph of rows of M. Proof. In BFS, each time a vertex v is dequeued, the vertices that are enqueued are those that have not previously been enqueued and that are neighbors of v. It suﬃces to show that this is what OverlapBFS does. L An invariant is that for each column c , the rows in are those rows that i i R have not been enqueued and whose left endpoints are in c , and the rows in i i are those that have not been enqueued and whose right endpoints are in c . This i R is true initially, and when a row is enqueued, it is removed from the two lists L L R i and or and that it occupies, maintaining the invariant. h (cid:48)i h A row R that has not been enqueued properly overlaps row R = (c , c , . . . , c ) j j+1 k if and only if one of of the following conditions applies: (cid:48) { } 1. The left endpoint of R is in c , c , . . . , c and its right endpoints is j+1 j+2 k to the right of c ; k ##LTFigure##

text: R

label: caption

text: Fig.

text: A

text: set

text: of

text: rows

text: of

text: a

text: consecutive-ones

text: matrix

text: whose

text: overlap

text: graph

text: Q

text: is

text: connected,

text: and

text: its

text: Venn

text: classes,

text: V

text: V

text: V

text: V

text: Each

text: element

text: of

text: U

text: V

text: V

text: V

text: is

text: consecutive,

text: their

text: union

text: is

text: consecutive,

text: and

text: the

text: order

text: of

text: V

text: R

text: through

text: V

text: is

text: uniquely

text: constrained

text: in

text: any

text: consecutive-ones

text: ordering

text: of

text: up

text: to

text: Q

text: reversal.

text: These

text: are

text: the

text: constrained

text: classes.

text: Columns

text: in

text: the

text: unconstrained

text: class,

text: V

text: can

text: go

text: on

text: either

text: end

text: of

text: this

text: sequence.

text: U

text: a

text: suﬃx

text: is

text: a

text: subpath

text: that

text: contains

text: its

text: last

text: row.

text: On

text: the

text: other

text: hand,

text: we

text: can

text: treat

text: P

text: it

text: as

text: an

text: unordered

text: collection

text: of

text: sets,

text: as

text: in

text: the

text: expression

text: Z

text: where

text: Z

text: is

text: a

text: P

text: row

text: not

text: in

text: Breadth-ﬁrst

text: search

text: on

text: the

text: overlap

text: graph

text: of

text: the

text: rows

text: of

text: a

text: matrix,

text: given

text: a

text: consecutive-ones

text: ordering

text: One

text: step

text: of

text: our

text: algorithm

text: is

text: to

text: ﬁnd

text: a

text: shortest

text: path

text: of

text: the

text: overlap

text: graph

text: of

text: the

text: rows

text: of

text: a

text: consecutive-ones

text: ordered

text: matrix.

text: The

text: diﬃculty

text: in

text: performing

text: BFS

text: on

text: the

text: overlap

text: graph

text: within

text: the

text: O(size(M))

text: time

text: bound

text: is

text: that

text: the

text: size

text: of

text: the

text: overlap

text: graph

text: is

text: not

text: O(size(M)).

text: A

text: simple

text: example

text: of

text: this

text: is

text: when

text: M

text: has

text: three

text: columns,

text: rows

text: with

text: in

text: the

text: ﬁrst

text: and

text: second

text: column

text: and

text: rows

text: with

text: in

text: the

text: second

text: and

text: third

text: column.

text: Then

text: size(M)

text: O(n)

text: but

text: its

text: overlap

text: graph

text: is

text: a

text: complete

text: bipartite

text: graph

text: with

text: edges.

text: The

text: algorithm

text: is

text: given

text: as

text: Algorithm

text: Lemma

text: The

text: parent

text: relation

text: assigned

text: by

text: OverlapBFS(M,

text: S)

text: (Algorithm

text: is

text: a

text: BFS

text: tree,

text: rooted

text: at

text: S,

text: in

text: the

text: overlap

text: graph

text: of

text: rows

text: of

text: M.

text: Proof.

text: In

text: BFS,

text: each

text: time

text: a

text: vertex

text: v

text: is

text: dequeued,

text: the

text: vertices

text: that

text: are

text: enqueued

text: are

text: those

text: that

text: have

text: not

text: previously

text: been

text: enqueued

text: and

text: that

text: are

text: neighbors

text: of

text: v.

text: It

text: suﬃces

text: to

text: show

text: that

text: this

text: is

text: what

text: OverlapBFS

text: does.

text: L

text: An

text: invariant

text: is

text: that

text: for

text: each

text: column

text: c

text: the

text: rows

text: in

text: are

text: those

text: rows

text: that

text: i

text: i

text: R

text: have

text: not

text: been

text: enqueued

text: and

text: whose

text: left

text: endpoints

text: are

text: in

text: c

text: and

text: the

text: rows

text: in

text: i

text: i

text: are

text: those

text: that

text: have

text: not

text: been

text: enqueued

text: and

text: whose

text: right

text: endpoints

text: are

text: in

text: c

text: This

text: i

text: R

text: is

text: true

text: initially,

text: and

text: when

text: a

text: row

text: is

text: enqueued,

text: it

text: is

text: removed

text: from

text: the

text: two

text: lists

text: L

text: L

text: R

text: i

text: and

text: or

text: and

text: that

text: it

text: occupies,

text: maintaining

text: the

text: invariant.

text: h

text: h

text: A

text: row

text: R

text: that

text: has

text: not

text: been

text: enqueued

text: properly

text: overlaps

text: row

text: R

text: (c

text: c

text: c

text: j

text: k

text: if

text: and

text: only

text: if

text: one

text: of

text: of

text: the

text: following

text: conditions

text: applies:

text: The

text: left

text: endpoint

text: of

text: R

text: is

text: in

text: c

text: c

text: c

text: and

text: its

text: right

text: endpoints

text: is

text: k

text: to

text: the

text: right

text: of

text: c

text: k

text: ##LTFigure##


text: (cid:16) (cid:17) (cid:16) (cid:17) T V ˙ ˙ Q, Q = Q, Q . (58) In other words, for the oscillatory eigenmodes there is an equipartition of the H T V T system energy, i.e., = + , between their kinetic energy and their potential V energy . 5 Spectral analysis of the system eigenmodes In this section we study time-harmonic solutions to the Euler-Lagrange equation (19) which constitutes a subject of the spectral theory. An important subject of this section is the study of the relations between standard spectral theory and the quadratic pencil formulation as it arises naturally as the time Fourier transformation of the Euler-Lagrange evolution equation. 5.1 Standard versus pencil formulations of the spectral prob- lems As introduced in Section 2.2, the eigenmodes of the Lagrangian system are nonzero − solutions of the ODEs (19) with F = 0 having the form Q (t) = qe iζt. For a ﬁxed β, these modes correspond to the solutions of the quadratic eigenvalue problem (QEP) (cid:54) C(ζ, β)q = 0, q = 0 (59) for the quadratic matrix pencil in ζ − C(ζ, β) = ζ2α + (2θ + βR) iζ η. (60) · The set of eigenvalues (spectrum) of the pencil C ( , β) is the set · { ∈ C } σ (C ( , β)) = ζ : det C(ζ, β) = 0 , (61) which are exactly those values ζ for which a solution to the QEP (59) exists. The spectral theory of polynomial operator pencils [Mar88] can be applied to study the eigenmodes but it has its disadvantages such as being more complicated than standard spectral theory. Thus an alternative approach to the spectral theory is desirable. Often the alternative approach is to use the Hamiltonian system and consider its eigenmodes, that is, the nonzero solutions of the Hamiltonian equations (27) with 17

label: equation

text: T

text: V

text: Q,

text: Q

text: Q,

text: Q

text: In

text: other

text: words,

text: for

text: the

text: oscillatory

text: eigenmodes

text: there

text: is

text: an

text: equipartition

text: of

text: the

text: H

text: T

text: V

text: T

text: system

text: energy,

text: i.e.,

text: between

text: their

text: kinetic

text: energy

text: and

text: their

text: potential

text: V

text: energy

label: section

text: Spectral

text: analysis

text: of

text: the

text: system

text: eigenmodes

text: In

text: this

text: section

text: we

text: study

text: time-harmonic

text: solutions

text: to

text: the

text: Euler-Lagrange

text: equation

text: which

text: constitutes

text: a

text: subject

text: of

text: the

text: spectral

text: theory.

text: An

text: important

text: subject

text: of

text: this

text: section

text: is

text: the

text: study

text: of

text: the

text: relations

text: between

text: standard

text: spectral

text: theory

text: and

text: the

text: quadratic

text: pencil

text: formulation

text: as

text: it

text: arises

text: naturally

text: as

text: the

text: time

text: Fourier

text: transformation

text: of

text: the

text: Euler-Lagrange

text: evolution

text: equation.

text: Standard

text: versus

text: pencil

text: formulations

text: of

text: the

text: spectral

text: prob-

text: lems

text: As

text: introduced

text: in

text: Section

text: the

text: eigenmodes

text: of

text: the

text: Lagrangian

text: system

text: are

text: nonzero

text: solutions

text: of

text: the

text: ODEs

text: with

text: F

text: having

text: the

text: form

text: Q

text: (t)

text: qe

text: iζt.

text: For

text: a

text: ﬁxed

text: β,

text: these

text: modes

text: correspond

text: to

text: the

text: solutions

text: of

text: the

text: quadratic

text: eigenvalue

text: problem

text: (QEP)

text: C(ζ,

text: β)q

text: q

text: for

text: the

text: quadratic

text: matrix

text: pencil

text: in

text: ζ

text: C(ζ,

text: β)

text: ζ2α

text: βR)

text: iζ

text: η.

text: The

text: set

text: of

text: eigenvalues

text: (spectrum)

text: of

text: the

text: pencil

text: C

text: β)

text: is

text: the

text: set

text: C

text: σ

text: (C

text: β))

text: ζ

text: det

text: C(ζ,

text: β)

text: which

text: are

text: exactly

text: those

text: values

text: ζ

text: for

text: which

text: a

text: solution

text: to

text: the

text: QEP

text: exists.

text: The

text: spectral

text: theory

text: of

text: polynomial

text: operator

text: pencils

text: can

text: be

text: applied

text: to

text: study

text: the

text: eigenmodes

text: but

text: it

text: has

text: its

text: disadvantages

text: such

text: as

text: being

text: more

text: complicated

text: than

text: standard

text: spectral

text: theory.

text: Thus

text: an

text: alternative

text: approach

text: to

text: the

text: spectral

text: theory

text: is

text: desirable.

text: Often

text: the

text: alternative

text: approach

text: is

text: to

text: use

text: the

text: Hamiltonian

text: system

text: and

text: consider

text: its

text: eigenmodes,

text: that

text: is,

text: the

text: nonzero

text: solutions

text: of

text: the

text: Hamiltonian

text: equations

text: with


text: − F = 0 having the form u (t) = ue iζt. These modes correspond to the solutions of the eigenvalue problem − (cid:54) Mu = iζu, u = 0 (62) (cid:18) (cid:20) (cid:21)(cid:19) for the matrix − βR 0 M (β) = J M . (63) 0 0 H (cid:20) (cid:21) An advantage to this approach is the simple correspondence via (7), (27) between the set of modes of the two systems, namely, the eigenmodes of the Hamiltonian p − system are solutions of (27) having the block form u (t) = e iζt in which Q (t) = q − − qe iζt an eigenmode of the Lagrangian system and p = ( iζα + θ) q. In particular, · this means the matrix iM (β) and the pencil C ( , β) have the same eigenvalues and hence the same spectrum, i.e., · σ (iM (β)) = σ (C ( , β)) . (64) A major disadvantage of this approach is that M (β) is a non-self-adjoint matrix such that the standard theory of self-adjoint or dissipative operators does not apply without further transformation of the system, even in the absence of losses, i.e., β = 0. § Instead, in this case it is the Krein spectral theory [Arnold, 42], [Yak], [YakSta] that is often used. But this theory is far more complex than the standard spectral theory and much harder to apply. Our approach to these spectral problems which overcomes the disadvantages in the pencil or Krein spectral theory is to use the canonical system and consider its eigenmodes, that is, the nonzero solutions of the canonical evolution equation (33) − with f = 0 having the form v (t) = we iζt. These modes correspond to the solutions of the eigenvalue problem (cid:54) A (β) w = ζw, w = 0 (65) − for the system operator A (β) = Ω iβB with Hermitian matrices Ω, B with ≥ B 0. The advantage of this approach is that the standard spectral theory can be − used since iA (β) is a dissipative operator and when losses are absent A(0) = Ω is self-adjoint. This is a serious advantage since the spectral theory is signiﬁcantly simpler and allows the usage of the deep and eﬀective results from perturbation theory such as those developed in [FigWel1] to study the modes of such canonical evolution equations having the general form (33). This is the approach we take in this paper 18

text: F

text: having

text: the

text: form

text: u

text: (t)

text: ue

text: iζt.

text: These

text: modes

text: correspond

text: to

text: the

text: solutions

text: of

text: the

text: eigenvalue

text: problem

font: default

text: Mu

text: iζu,

text: u

text: for

text: the

text: matrix

text: βR

text: M

text: (β)

text: J

text: M

text: H

text: An

text: advantage

text: to

text: this

text: approach

text: is

text: the

text: simple

text: correspondence

text: via

text: between

text: the

text: set

text: of

text: modes

text: of

text: the

text: two

text: systems,

text: namely,

text: the

text: eigenmodes

text: of

text: the

text: Hamiltonian

text: p

text: system

text: are

text: solutions

text: of

text: having

text: the

text: block

text: form

text: u

text: (t)

text: e

text: iζt

text: in

text: which

text: Q

text: (t)

text: q

text: qe

text: iζt

text: an

text: eigenmode

text: of

text: the

text: Lagrangian

text: system

text: and

text: p

text: iζα

text: θ)

text: q.

text: In

text: particular,

text: this

text: means

text: the

text: matrix

text: iM

text: (β)

text: and

text: the

text: pencil

text: C

text: β)

text: have

text: the

text: same

text: eigenvalues

text: and

text: hence

text: the

text: same

text: spectrum,

text: i.e.,

text: σ

text: (iM

text: (β))

text: σ

text: (C

text: β))

text: A

text: major

text: disadvantage

text: of

text: this

text: approach

text: is

text: that

text: M

text: (β)

text: is

text: a

text: non-self-adjoint

text: matrix

text: such

text: that

text: the

text: standard

text: theory

text: of

text: self-adjoint

text: or

text: dissipative

text: operators

text: does

text: not

text: apply

text: without

text: further

text: transformation

text: of

text: the

text: system,

text: even

text: in

text: the

text: absence

text: of

text: losses,

text: i.e.,

text: β

text: Instead,

text: in

text: this

text: case

text: it

text: is

text: the

text: Krein

text: spectral

text: theory

text: [Arnold,

text: [Yak],

text: [YakSta]

text: that

text: is

text: often

text: used.

text: But

text: this

text: theory

text: is

text: far

text: more

text: complex

text: than

text: the

text: standard

text: spectral

text: theory

text: and

text: much

text: harder

text: to

text: apply.

text: Our

text: approach

text: to

text: these

text: spectral

text: problems

text: which

text: overcomes

text: the

text: disadvantages

text: in

text: the

text: pencil

text: or

text: Krein

text: spectral

text: theory

text: is

text: to

text: use

text: the

text: canonical

text: system

text: and

text: consider

text: its

text: eigenmodes,

text: that

text: is,

text: the

text: nonzero

text: solutions

text: of

text: the

text: canonical

text: evolution

text: equation

text: with

text: f

text: having

text: the

text: form

text: v

text: (t)

text: we

text: iζt.

text: These

text: modes

text: correspond

text: to

text: the

text: solutions

text: of

text: the

text: eigenvalue

text: problem

text: A

text: (β)

text: w

text: ζw,

text: w

text: for

text: the

text: system

text: operator

text: A

text: (β)

text: Ω

text: iβB

text: with

text: Hermitian

text: matrices

text: Ω,

text: B

text: with

text: B

text: The

text: advantage

text: of

text: this

text: approach

text: is

text: that

text: the

text: standard

text: spectral

text: theory

text: can

text: be

text: used

text: since

text: iA

text: (β)

text: is

text: a

text: dissipative

text: operator

text: and

text: when

text: losses

text: are

text: absent

text: Ω

text: is

text: self-adjoint.

text: This

text: is

text: a

text: serious

text: advantage

text: since

text: the

text: spectral

text: theory

text: is

text: signiﬁcantly

text: simpler

text: and

text: allows

text: the

text: usage

text: of

text: the

text: deep

text: and

text: eﬀective

text: results

text: from

text: perturbation

text: theory

text: such

text: as

text: those

text: developed

text: in

text: to

text: study

text: the

text: modes

text: of

text: such

text: canonical

text: evolution

text: equations

text: having

text: the

text: general

text: form

text: This

text: is

text: the

text: approach

text: we

text: take

text: in

text: this

text: paper


text: Theorem 9 (modal dichotomy I) If β > 2ωmax then bmin ∪ ∩ ∅ σ (A (β)) = σ (A (β)) σ (A (β)) , σ (A (β)) σ (A (β)) = . (85) 0 1 0 1 Furthermore, there exists unique invariant subspaces H (β), H (β) of the system − (cid:96)(cid:96) h(cid:96) operator A (β) = Ω iβB with the properties (cid:0) (cid:1) (cid:0) (cid:1) ⊕ (i) H = H (β) H (β) ; (86) (cid:96)(cid:96) h(cid:96) | | (ii) σ A (β) = σ (A (β)) , σ A (β) = σ (A (β)) , H (β) 0 H (β) 1 (cid:96)(cid:96) h(cid:96) C where H = 2N. Moreover, the dimensions of these subspaces satisfy − dim H (β) = N , dim H (β) = 2N N . (87) h(cid:96) R (cid:96)(cid:96) R Deﬁnition 10 (high-loss susceptible subspace) For the system operator A (β) = − Ω iβB with β > 2ωmax we will call its N -dimensional invariant subspace H (β) bmin R − h(cid:96) the high-loss susceptible subspace. We will call its (2N N )-dimensional invariant R subspace H (β) the low-loss susceptible subspace. (cid:96)(cid:96) Our reasoning for the deﬁnitions of these subspaces is clariﬁed with the following corollary. Corollary (cid:0)11 (high-lo(cid:1)ss subspace: dissipative properties) If β > 2ωmax then bmin (cid:0) (cid:1) | { ∈ ≤ − ≤ } σ A (β) = ζ σ (A (β)) : 0 Im ζ ω , (88) H (β) max | (cid:96)(cid:96) { ∈ − ≥ − } σ A (β) = ζ σ (A (β)) : Im ζ βb ω > ω . H (β) min max max h(cid:96) Furthermore, the quality factor (43) of any eigenmode of the canonical system (33) in the high-loss susceptible subspace H (β) satisﬁes h(cid:96) ≤ ≤ 1 ω 1 0 max Q[w] m−ax < . (89) w an eigenvector 2 βbmin ωmax 2 of A(β) in H (β) h(cid:96) ∞ In particular, as the losses go to the damping factor and quality factor of any ∞ such eigenmode goes to + and 0, respectively, that is, − ∞ lim min ( Im ζ) = + , lim max Q[w] = 0. (90) →∞ ∈ | →∞ β ζ σ(A(β) ) β w an eigenvector Hh(cid:96)(β) of A(β) in H (β) h(cid:96) 23 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Theorem

text: (modal

text: dichotomy

text: I)

text: If

text: β

text: then

text: bmin

text: σ

text: (A

text: (β))

text: σ

text: (A

text: (β))

text: σ

text: (A

text: (β))

text: σ

text: (A

text: (β))

text: σ

text: (A

text: (β))

text: Furthermore,

text: there

text: exists

text: unique

text: invariant

text: subspaces

text: H

text: (β),

text: H

text: (β)

text: of

text: the

text: system

text: operator

text: A

text: (β)

text: Ω

text: iβB

text: with

text: the

text: properties

text: (i)

text: H

text: H

text: (β)

text: H

text: (β)

text: (ii)

text: σ

text: A

text: (β)

text: σ

text: (A

text: (β))

text: σ

text: A

text: (β)

text: σ

text: (A

text: (β))

text: H

text: (β)

text: H

text: (β)

text: C

text: where

text: H

text: Moreover,

text: the

text: dimensions

text: of

text: these

text: subspaces

text: satisfy

text: dim

text: H

text: (β)

text: N

text: dim

text: H

text: (β)

text: N

text: R

text: R

text: Deﬁnition

text: (high-loss

text: susceptible

text: subspace)

text: For

text: the

text: system

text: operator

text: A

text: (β)

text: Ω

text: iβB

text: with

text: β

text: we

text: will

text: call

text: its

text: N

text: -dimensional

text: invariant

text: subspace

text: H

text: (β)

text: bmin

text: R

text: the

text: high-loss

text: susceptible

text: subspace.

text: We

text: will

text: call

text: its

text: N

text: )-dimensional

text: invariant

text: R

text: subspace

text: H

text: (β)

text: the

text: low-loss

text: susceptible

text: subspace.

text: Our

text: reasoning

text: for

text: the

text: deﬁnitions

text: of

text: these

text: subspaces

text: is

text: clariﬁed

text: with

text: the

text: following

text: corollary.

text: Corollary

text: subspace:

text: dissipative

text: properties)

text: If

text: β

text: then

text: bmin

text: σ

text: A

text: (β)

text: ζ

text: σ

text: (A

text: (β))

text: Im

text: ζ

text: ω

text: H

text: (β)

text: max

text: σ

text: A

text: (β)

text: ζ

text: σ

text: (A

text: (β))

text: Im

text: ζ

text: βb

text: ω

text: ω

text: H

text: (β)

text: min

text: max

text: max

text: Furthermore,

text: the

text: quality

text: factor

text: of

text: any

text: eigenmode

text: of

text: the

text: canonical

text: system

text: in

text: the

text: high-loss

text: susceptible

text: subspace

text: H

text: (β)

text: satisﬁes

text: ω

text: max

text: Q[w]

text: m−ax

text: w

text: an

text: eigenvector

text: βbmin

text: ωmax

text: of

text: A(β)

text: in

text: H

text: (β)

text: In

text: particular,

text: as

text: the

text: losses

text: go

text: to

text: the

text: damping

text: factor

text: and

text: quality

text: factor

text: of

text: any

text: such

text: eigenmode

text: goes

text: to

text: and

text: respectively,

text: that

text: is,

text: lim

text: min

text: Im

text: ζ)

text: lim

text: max

text: Q[w]

text: β

text: ζ

text: σ(A(β)

text: β

text: w

text: an

text: eigenvector

text: of

text: A(β)

text: in

text: H

text: (β)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:0) (cid:1) We conclude this section with the following remarks. The spectrum of A (β) | restricted to the low-loss susceptible subspace H (β), that is, the set σ A (β) (cid:96)(cid:96) H (β) (cid:96)(cid:96) in (88), is close to the real axis and actually coalesces to a ﬁnite set of real numbers as → ∞ losses β . This statement is made precise in the following section with Theorem 12 and the asymptotic expansions in (99). On the other hand, results on quality factor for the eigenmodes in H (β) is far more subtle than the results in Corollary 11 for the (cid:96)(cid:96) eigenmodes in the high-loss susceptible subspace H (β). For gyroscopic-dissipative h(cid:96) systems considered in this paper, Corollary 11 above and Proposition 13 below give (cid:29) a partial description of the nature of the quality factor for large losses, i.e., β 1, in the lossy component of the composite system. When gyroscopy is absent, i.e., θ = 0, then a more complete analysis for quality factor can be carried out, which we have done in Section 6 of this paper in connection to our studies on the overdamping phenomenon. Consideration of quality factor and overdamping in dissipative systems with gyroscopy, however, requires a more subtle and detailed analysis that will be considered in a future work. 5.2.3 Modal dichotomy in the high-loss regime We are interested in describing the spectrum σ (A (β)) of the system operator − ≥ (cid:29) A (β) = Ω iβB, β 0 in the high-loss regime, i.e., β 1. We do this in this → ∞ section by giving an asymptotic characterization, as β , of the modal dichotomy as described in Theorem 9 and Corollary 11. In order to do so we need to give → ∞ a spectral perturbation analysis of the matrix A (β) as β . Fortunately, this analysis has already been carried out in [FigWel1]. We now introduce the necessary notion and describe the results. C · · The Hilbert space H = 2N with standard inner product ( , ) is decomposed into the direct sum of orthogonal invariant subspaces of the operator B, namely ⊕ ⊥ H = H H , dim H = N , (91) B B B R where H = Ran B (the range of B) is the loss subspace of dimension N = B ⊥ R rank B with orthogonal projection P and its orthogonal complement, H = Ker B B − B (the nullspace of B), is the no-loss subspace of dimension 2N N with orthogonal ⊥ R projection P . B × The operators Ω and B w(cid:20)ith respec(cid:21)t to the d(cid:20)irect sum(cid:21) (91) are the 2 2 block operator matrices Ω Θ B 0 Ω = 2∗ , B = 2 , (92) Θ Ω 0 0 1 | → | → where Ω = P ΩP : H H and B = P BP : H H are 2 B B HB B B 2 B B HB B B restrictions of the operators Ω and B respectively to loss subspace H whereas Ω = B 1 24

text: We

text: conclude

text: this

text: section

text: with

text: the

text: following

text: remarks.

text: The

text: spectrum

text: of

text: A

text: (β)

text: restricted

text: to

text: the

text: low-loss

text: susceptible

text: subspace

text: H

text: (β),

text: that

text: is,

text: the

text: set

text: σ

text: A

text: (β)

text: H

text: (β)

text: in

text: is

text: close

text: to

text: the

text: real

text: axis

text: and

text: actually

text: coalesces

text: to

text: a

text: ﬁnite

text: set

text: of

text: real

text: numbers

text: as

text: losses

text: β

text: This

text: statement

text: is

text: made

text: precise

text: in

text: the

text: following

text: section

text: with

text: Theorem

text: and

text: the

text: asymptotic

text: expansions

text: in

text: On

text: the

text: other

text: hand,

text: results

text: on

text: quality

text: factor

text: for

text: the

text: eigenmodes

text: in

text: H

text: (β)

text: is

text: far

text: more

text: subtle

text: than

text: the

text: results

text: in

text: Corollary

text: for

text: the

text: eigenmodes

text: in

text: the

text: high-loss

text: susceptible

text: subspace

text: H

text: (β).

text: For

text: gyroscopic-dissipative

text: systems

text: considered

text: in

text: this

text: paper,

text: Corollary

text: above

text: and

text: Proposition

text: below

text: give

text: a

text: partial

text: description

text: of

text: the

text: nature

text: of

text: the

text: quality

text: factor

text: for

text: large

text: losses,

text: i.e.,

text: β

text: in

text: the

text: lossy

text: component

text: of

text: the

text: composite

text: system.

text: When

text: gyroscopy

text: is

text: absent,

text: i.e.,

text: θ

text: then

text: a

text: more

text: complete

text: analysis

text: for

text: quality

text: factor

text: can

text: be

text: carried

text: out,

text: which

text: we

text: have

text: done

text: in

text: Section

text: of

text: this

text: paper

text: in

text: connection

text: to

text: our

text: studies

text: on

text: the

text: overdamping

text: phenomenon.

text: Consideration

text: of

text: quality

text: factor

text: and

text: overdamping

text: in

text: dissipative

text: systems

text: with

text: gyroscopy,

text: however,

text: requires

text: a

text: more

text: subtle

text: and

text: detailed

text: analysis

text: that

text: will

text: be

text: considered

text: in

text: a

text: future

text: work.

text: Modal

text: dichotomy

text: in

text: the

text: high-loss

text: regime

text: We

text: are

text: interested

text: in

text: describing

text: the

text: spectrum

text: σ

text: (A

text: (β))

text: of

text: the

text: system

text: operator

text: A

text: (β)

text: Ω

text: iβB,

text: β

text: in

text: the

text: high-loss

text: regime,

text: i.e.,

text: β

text: We

text: do

text: this

text: in

text: this

text: section

text: by

text: giving

text: an

text: asymptotic

text: characterization,

text: as

text: β

text: of

text: the

text: modal

text: dichotomy

text: as

text: described

text: in

text: Theorem

text: and

text: Corollary

text: In

text: order

text: to

text: do

text: so

text: we

text: need

text: to

text: give

text: a

text: spectral

text: perturbation

text: analysis

text: of

text: the

text: matrix

text: A

text: (β)

text: as

text: β

text: Fortunately,

text: this

text: analysis

text: has

text: already

text: been

text: carried

text: out

text: in

text: We

text: now

text: introduce

text: the

text: necessary

text: notion

text: and

text: describe

text: the

text: results.

text: C

text: The

text: Hilbert

text: space

text: H

text: with

text: standard

text: inner

text: product

text: is

text: decomposed

text: into

text: the

text: direct

text: sum

text: of

text: orthogonal

text: invariant

text: subspaces

text: of

text: the

text: operator

text: B,

text: namely

text: H

text: H

text: H

text: dim

text: H

text: N

text: B

text: B

text: B

text: R

text: where

text: H

text: Ran

text: B

text: (the

text: range

text: of

text: B)

text: is

text: the

text: loss

text: subspace

text: of

text: dimension

text: N

text: B

text: R

text: rank

text: B

text: with

text: orthogonal

text: projection

text: P

text: and

text: its

text: orthogonal

text: complement,

text: H

text: Ker

text: B

text: B

text: B

text: (the

text: nullspace

text: of

text: B),

text: is

text: the

text: no-loss

text: subspace

text: of

text: dimension

text: N

text: with

text: orthogonal

text: R

text: projection

text: P

text: B

text: The

text: operators

text: Ω

text: and

text: B

text: to

text: the

text: are

text: the

text: block

text: operator

text: matrices

text: Ω

text: Θ

text: B

text: Ω

text: 2∗

text: B

text: Θ

text: Ω

text: where

text: Ω

text: P

text: ΩP

text: H

text: H

text: and

text: B

text: P

text: BP

text: H

text: H

text: are

text: B

text: B

text: HB

text: B

text: B

text: B

text: B

text: HB

text: B

text: B

text: restrictions

text: of

text: the

text: operators

text: Ω

text: and

text: B

text: respectively

text: to

text: loss

text: subspace

text: H

text: whereas

text: Ω

text: B


text: (cid:54) − ρ = 0 then the limiting function lim →∞ v (t, β) = w˚ e iρjt (for ﬁx t) is an oscillatory j β j → ∞ j function in t with period 2π/ρ . Thus as β we denote this by j ∼ − (cid:54) asymptotic oscillatory low-loss modes: v (t, β) w˚ e iρjt, ρ = 0. (112) j j j From the quality factor discussion above for the low-loss modes the next proposi- (cid:54) tion follows immediately, regardless of whether d = 0 or not. j Proposition 13 (Q-factor: asymptotic oscillatory low-loss modes) Any low- (cid:54) loss eigenmodes from (105) with ρ = 0 has a quality factor Q [w (β)] with the prop- ∞ j (cid:29) (cid:37) j∞ → ∞ erty that either Q [w (β)] = + for all β 1 or Q [w (β)] + as β (i.e., j j is an unbounded increasing function of the loss parameter β). 6 Overdamping analysis The phenomenon of overdamping (also called heavy damping) is best known for a simple damped oscillator. Namely, when the damping exceeds certain critical value all oscillations cease entirely, see, for instance, [Pain, Sec. 2]. In other words, if the − damped oscillations are described by the exponential function e iζt with a complex constant ζ then in the case of overdamping (heavy damping) Re ζ = 0. Our interest to overdamping is motivated by the fact that if an eigenmode becomes overdamped it can not resonate at any ﬁnite frequency. Consequently, the contribution of such a mode to losses at ﬁnite frequencies becomes minimal, and that provides a mechanism for the absorption suppression for systems composed of lossy and lossless components. The treatment of overdamping for systems with many degrees of freedom involves a number of subtleties particularly in our case when the both lossy and lossless degrees of freedom are present, i.e., when the loss fraction condition (23) is satisﬁed. We will show that any Lagrangian system (19) with θ = 0 can be overdamped with selective overdamping occurring whenever R does not have full rank, i.e., N < N, which R corresponds to a model of a two-component composite with a high-loss and a low-loss component. Let us be clear that we study overdamping in this paper only for the case θ = 0 (cid:54) as the case θ = 0 will diﬀer signiﬁcantly in the analysis and it is not expected that there exists such a critical value for damping when all oscillations cease entirely for any damped oscillation. A study of a proper generalization of overdamping in the (cid:54) case θ = 0 will be carried out in a future publication. We make here statements and provide arguments for overdamping for Lagrangian systems. We will study the phenomenon known as overdamping in this section under the following condition: 29

text: ρ

text: then

text: the

text: limiting

text: function

text: lim

text: v

text: (t,

text: β)

text: w˚

text: e

text: iρjt

text: (for

text: ﬁx

text: t)

text: is

text: an

text: oscillatory

text: j

text: β

text: j

text: j

text: function

text: in

text: t

text: with

text: period

text: 2π/ρ

text: Thus

text: as

text: β

text: we

text: denote

text: this

text: by

text: j

text: asymptotic

text: oscillatory

text: low-loss

text: modes:

text: v

text: (t,

text: β)

text: w˚

text: e

text: iρjt,

text: ρ

text: j

text: j

text: j

text: From

text: the

text: quality

text: factor

text: discussion

text: above

text: for

text: the

text: low-loss

text: modes

text: the

text: next

text: proposi-

text: tion

text: follows

text: immediately,

text: regardless

text: of

text: whether

text: d

text: or

text: not.

text: j

text: Proposition

text: (Q-factor:

text: asymptotic

text: oscillatory

text: low-loss

text: modes)

text: Any

text: low-

text: loss

text: eigenmodes

text: from

text: with

text: ρ

text: has

text: a

text: quality

text: factor

text: Q

text: [w

text: (β)]

text: with

text: the

text: prop-

text: j

text: j∞

text: erty

text: that

text: either

text: Q

text: [w

text: (β)]

text: for

text: all

text: β

text: or

text: Q

text: [w

text: (β)]

text: as

text: β

text: (i.e.,

text: j

text: j

text: is

text: an

text: unbounded

text: increasing

text: function

text: of

text: the

text: loss

text: parameter

text: β).

text: Overdamping

text: analysis

text: The

text: phenomenon

text: of

text: overdamping

text: (also

text: called

text: heavy

text: damping)

text: is

text: best

text: known

text: for

text: a

text: simple

text: damped

text: oscillator.

text: Namely,

text: when

text: the

text: damping

text: exceeds

text: certain

text: critical

text: value

text: all

text: oscillations

text: cease

text: entirely,

text: see,

text: for

text: instance,

text: [Pain,

text: Sec.

text: In

text: other

text: words,

text: if

text: the

text: damped

text: oscillations

text: are

text: described

text: by

text: the

text: exponential

text: function

text: e

text: iζt

text: with

text: a

text: complex

text: constant

text: ζ

text: then

text: in

text: the

text: case

text: of

text: overdamping

text: (heavy

text: damping)

text: Re

text: ζ

text: Our

text: interest

text: to

text: overdamping

text: is

text: motivated

text: by

text: the

text: fact

text: that

text: if

text: an

text: eigenmode

text: becomes

text: overdamped

text: it

text: can

text: not

text: resonate

text: at

text: any

text: ﬁnite

text: frequency.

text: Consequently,

text: the

text: contribution

text: of

text: such

text: a

text: mode

text: to

text: losses

text: at

text: ﬁnite

text: frequencies

text: becomes

text: minimal,

text: and

text: that

text: provides

text: a

text: mechanism

text: for

text: the

text: absorption

text: suppression

text: for

text: systems

text: composed

text: of

text: lossy

text: and

text: lossless

text: components.

text: The

text: treatment

text: of

text: overdamping

text: for

text: systems

text: with

text: many

text: degrees

text: of

text: freedom

text: involves

text: a

text: number

text: of

text: subtleties

text: particularly

text: in

text: our

text: case

text: when

text: the

text: both

text: lossy

text: and

text: lossless

text: degrees

text: of

text: freedom

text: are

text: present,

text: i.e.,

text: when

text: the

text: loss

text: fraction

text: condition

text: is

text: satisﬁed.

text: We

text: will

text: show

text: that

text: any

text: Lagrangian

text: system

text: with

text: θ

text: can

text: be

text: overdamped

text: with

text: selective

text: overdamping

text: occurring

text: whenever

text: R

text: does

text: not

text: have

text: full

text: rank,

text: i.e.,

text: N

text: N,

text: which

text: R

text: corresponds

text: to

text: a

text: model

text: of

text: a

text: two-component

text: composite

text: with

text: a

text: high-loss

text: and

text: a

text: low-loss

text: component.

text: Let

text: us

text: be

text: clear

text: that

text: we

text: study

text: overdamping

text: in

text: this

text: paper

text: only

text: for

text: the

text: case

text: θ

text: as

text: the

text: case

text: θ

text: will

text: diﬀer

text: signiﬁcantly

text: in

text: the

text: analysis

text: and

text: it

text: is

text: not

text: expected

text: that

text: there

text: exists

text: such

text: a

text: critical

text: value

text: for

text: damping

text: when

text: all

text: oscillations

text: cease

text: entirely

text: for

text: any

text: damped

text: oscillation.

text: A

text: study

text: of

text: a

text: proper

text: generalization

text: of

text: overdamping

text: in

text: the

text: case

text: θ

text: will

text: be

text: carried

text: out

text: in

text: a

text: future

text: publication.

text: We

text: make

text: here

text: statements

text: and

text: provide

text: arguments

text: for

text: overdamping

text: for

text: Lagrangian

text: systems.

text: We

text: will

text: study

text: the

text: phenomenon

text: known

text: as

text: overdamping

text: in

text: this

text: section

text: under

text: the

text: following

text: condition:


text: In fact, we show that the fraction of all overdamped modes is exactly the loss fraction δ which satisﬁes 0 < δ < 1, and half of these overdamped modes are R R − the high-loss modes. Thus it is exactly the positive fraction 1 δ > 0 of modes R which are low-loss oscillatory modes. In addition, these latter modes have extremely high quality factor which increases as the losses in the lossy component increase. An example of this behavior using the electric circuit in Section 3 was shown numerically in [FigWel1]. Virial theorem for dissipative systems and equipartition of energy. Re- § call that the virial theorem from classical mechanics [Gold, pp. 83-86; 3.4] (which we review in Appendix 11) is about the oscillatory transfer of energy from one form to another and its equipartition. For conservative linear systems whose Lagrangian is the diﬀerence between the kinetic and potential energy, e.g., a spring-mass system or a parallel LC circuit, the virial theorem says that for any time-periodic state of the system, the time-averaged kinetic energy equals the time-averaged potential en- ergy. This result is known as the equipartition of energy because of the fact that the total energy of the system, i.e., the Hamiltonian, is equal to the sum of the kinetic and potential energy. It is this result that we generalize in this paper for dissipative Lagrangian systems and their damped oscillatory modes. In particular, we show that for any damped oscillatory mode of the system the kinetic energy equals the potential energy and so there is an equipartition of the system energy for these damped oscillatory modes into kinetic and potential energy. Our precise statements of these results can be found in Section 4 in Theorem 1 and Corollary 2. We have also found that the transition to overdamping is characterized by a break- down of the virial theorem. More speciﬁcally, as the losses of the lossy component increase some of the modes become overdamped with complete ceasing of any oscil- lations and with breaking of the equality between kinetic and potential energy. Such a breaking of the equipartition of the energy can viewed as a dynamical characteri- zation of the overdamped modes! So we now have two diﬀerent characterizations of overdamping: spectral and dynamical. The eﬀectiveness of this dynamical character- ization is demonstrated in our study of the selective overdamping phenomenon and in proving that a positive fraction of modes always remain oscillatory no matter how large the losses become in the lossy component of the composite system. Standard spectral theory vs. Krein spectral theory. The study of the eigenmodes of a Lagrangian system often relies on the fact the system evolution can be transformed into the Hamiltonian form as the ﬁrst-order linear diﬀerential equations. However, it is not emphasized enough that the Hamiltonian setting does not lead directly to the standard spectral theory of self-adjoint or dissipative operators but that only it might be reduced to one in some important and well known cases by a proper transformation as, for instance, in the case of a simple oscillator. The general spectral theory of Hamiltonian systems is known as the Krein spectral § theory [Arnold, 42], [Yak], [YakSta]. This theory is far more complex than the 4

text: In

text: fact,

text: we

text: show

text: that

text: the

text: fraction

text: of

text: all

text: overdamped

text: modes

text: is

text: exactly

text: the

text: loss

text: fraction

text: δ

text: which

text: satisﬁes

text: δ

text: and

text: half

text: of

text: these

text: overdamped

text: modes

text: are

text: R

text: R

text: the

text: high-loss

text: modes.

text: Thus

text: it

text: is

text: exactly

text: the

text: positive

text: fraction

text: δ

text: of

text: modes

text: R

text: which

text: are

text: low-loss

text: oscillatory

text: modes.

text: In

text: addition,

text: these

text: latter

text: modes

text: have

text: extremely

text: high

text: quality

text: factor

text: which

text: increases

text: as

text: the

text: losses

text: in

text: the

text: lossy

text: component

text: increase.

text: An

text: example

text: of

text: this

text: behavior

text: using

text: the

text: electric

text: circuit

text: in

text: Section

text: was

text: shown

text: numerically

text: in

text: Virial

text: theorem

text: for

text: dissipative

text: systems

text: and

text: equipartition

text: of

text: energy.

text: Re-

text: call

text: that

text: the

text: virial

text: theorem

text: from

text: classical

text: mechanics

text: [Gold,

text: pp.

text: (which

text: we

text: review

text: in

text: Appendix

text: is

text: about

text: the

text: oscillatory

text: transfer

text: of

text: energy

text: from

text: one

text: form

text: to

text: another

text: and

text: its

text: equipartition.

text: For

text: conservative

text: linear

text: systems

text: whose

text: Lagrangian

text: is

text: the

text: diﬀerence

text: between

text: the

text: kinetic

text: and

text: potential

text: energy,

text: e.g.,

text: a

text: spring-mass

text: system

text: or

text: a

text: parallel

text: LC

text: circuit,

text: the

text: virial

text: theorem

text: says

text: that

text: for

text: any

text: time-periodic

text: state

text: of

text: the

text: system,

text: the

text: time-averaged

text: kinetic

text: energy

text: equals

text: the

text: time-averaged

text: potential

text: en-

text: ergy.

text: This

text: result

text: is

text: known

text: as

text: the

text: equipartition

text: of

text: energy

text: because

text: of

text: the

text: fact

text: that

text: the

text: total

text: energy

text: of

text: the

text: system,

text: i.e.,

text: the

text: Hamiltonian,

text: is

text: equal

text: to

text: the

text: sum

text: of

text: the

text: kinetic

text: and

text: potential

text: energy.

text: It

text: is

text: this

text: result

text: that

text: we

text: generalize

text: in

text: this

text: paper

text: for

text: dissipative

text: Lagrangian

text: systems

text: and

text: their

text: damped

text: oscillatory

text: modes.

text: In

text: particular,

text: we

text: show

text: that

text: for

text: any

text: damped

text: oscillatory

text: mode

text: of

text: the

text: system

text: the

text: kinetic

text: energy

text: equals

text: the

text: potential

text: energy

text: and

text: so

text: there

text: is

text: an

text: equipartition

text: of

text: the

text: system

text: energy

text: for

text: these

text: damped

text: oscillatory

text: modes

text: into

text: kinetic

text: and

text: potential

text: energy.

text: Our

text: precise

text: statements

text: of

text: these

text: results

text: can

text: be

text: found

text: in

text: Section

text: in

text: Theorem

text: and

text: Corollary

text: We

text: have

text: also

text: found

text: that

text: the

text: transition

text: to

text: overdamping

text: is

text: characterized

text: by

text: a

text: break-

text: down

text: of

text: the

text: virial

text: theorem.

text: More

text: speciﬁcally,

text: as

text: the

text: losses

text: of

text: the

text: lossy

text: component

text: increase

text: some

text: of

text: the

text: modes

text: become

text: overdamped

text: with

text: complete

text: ceasing

text: of

text: any

text: oscil-

text: lations

text: and

text: with

text: breaking

text: of

text: the

text: equality

text: between

text: kinetic

text: and

text: potential

text: energy.

text: Such

text: a

text: breaking

text: of

text: the

text: equipartition

text: of

text: the

text: energy

text: can

text: viewed

text: as

text: a

text: dynamical

text: characteri-

text: zation

text: of

text: the

text: overdamped

text: modes!

text: So

text: we

text: now

text: have

text: two

text: diﬀerent

text: characterizations

text: of

text: overdamping:

text: spectral

text: and

text: dynamical.

text: The

text: eﬀectiveness

text: of

text: this

text: dynamical

text: character-

text: ization

text: is

text: demonstrated

text: in

text: our

text: study

text: of

text: the

text: selective

text: overdamping

text: phenomenon

text: and

text: in

text: proving

text: that

text: a

text: positive

text: fraction

text: of

text: modes

text: always

text: remain

text: oscillatory

text: no

text: matter

text: how

text: large

text: the

text: losses

text: become

text: in

text: the

text: lossy

text: component

text: of

text: the

text: composite

text: system.

text: Standard

text: spectral

text: theory

text: vs.

text: Krein

text: spectral

text: theory.

text: The

text: study

text: of

text: the

text: eigenmodes

text: of

text: a

text: Lagrangian

text: system

text: often

text: relies

text: on

text: the

text: fact

text: the

text: system

text: evolution

text: can

text: be

text: transformed

text: into

text: the

text: Hamiltonian

text: form

text: as

text: the

text: ﬁrst-order

text: linear

text: diﬀerential

text: equations.

text: However,

text: it

text: is

text: not

text: emphasized

text: enough

text: that

text: the

text: Hamiltonian

text: setting

text: does

text: not

text: lead

text: directly

text: to

text: the

text: standard

text: spectral

text: theory

text: of

text: self-adjoint

text: or

text: dissipative

text: operators

text: but

text: that

text: only

text: it

text: might

text: be

text: reduced

text: to

text: one

text: in

text: some

text: important

text: and

text: well

text: known

text: cases

text: by

text: a

text: proper

text: transformation

text: as,

text: for

text: instance,

text: in

text: the

text: case

text: of

text: a

text: simple

text: oscillator.

text: The

text: general

text: spectral

text: theory

text: of

text: Hamiltonian

text: systems

text: is

text: known

text: as

text: the

text: Krein

text: spectral

text: theory

text: [Arnold,

text: [Yak],

text: [YakSta].

text: This

text: theory

text: is

text: far

text: more

text: complex

text: than

text: the


text: − ∈ R det C (ζ, β) = 0, ζ = iλ, λ . (118) Thus to study overdamping it suﬃces to study conditions under which the quadratic eigenvalue problem − (cid:54) ∈ R C ( iλ, β) q = 0, q = 0, λ . (119) (cid:0) (cid:1) has a solution. To be explicit we have − − − C ( iλ, β) = λ2α λβR + η . (120) As it turns out the key to studying overdamping will be to study zero sets of the family of quadratic forms − − − q (q, λ) = (q, C ( iλ, β) q) = λ2 (q, αq) λβ (q, Rq) + (q, ηq) (121) C on certain subspaces of the Hilbert space H = N. In particular, we ﬁnd that p (cid:115) (cid:20) (cid:21) ∈ − (cid:54) ⇒ q Ker C ( iλ, β) , q = 0 q (q, λ) = 0 and (122) β (q, Rq) ± β (q, Rq) 2 − (q, ηq) λ = . 2 (q, αq) 2 (q, αq) (q, αq) Thus we ﬁnd that a necessary and suﬃcient condition for an eigenvalue ζ of the (cid:115) system operator A (β) to satisfy Re ζ = 0 is β (q, Rq) ≥ (q, ηq) ∈ (cid:54) , for some q Ker C (ζ, β) , q = 0. (123) 2 (q, αq) (q, αq) Now there are some fundamental inequalities that play a key role in our over- damping analysis which are described in the following proposition. We will prove part of this proposition, the rest is proved in Section 7, since the proof is enlightening showing how the energy conservation law (20) and the virial theorem 1 can be used to derive these inequalities. Proposition 16 (fundamental inequalities) The following statements are true. ∈ C (cid:54) 1. For any q N with q = 0, 31 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: R

text: det

text: C

text: (ζ,

text: β)

text: ζ

text: iλ,

text: λ

text: Thus

text: to

text: study

text: overdamping

text: it

text: suﬃces

text: to

text: study

text: conditions

text: under

text: which

text: the

text: quadratic

text: eigenvalue

text: problem

text: R

text: C

text: iλ,

text: β)

text: q

text: q

text: λ

text: has

text: a

text: solution.

text: To

text: be

text: explicit

text: we

text: have

text: C

text: iλ,

text: β)

text: λ2α

text: λβR

text: η

text: As

text: it

text: turns

text: out

text: the

text: key

text: to

text: studying

text: overdamping

text: will

text: be

text: to

text: study

text: zero

text: sets

text: of

text: the

text: family

text: of

text: quadratic

text: forms

text: q

text: (q,

text: λ)

text: (q,

text: C

text: iλ,

text: β)

text: q)

text: λ2

text: (q,

text: αq)

text: λβ

text: (q,

text: Rq)

text: (q,

text: ηq)

text: C

text: on

text: certain

text: subspaces

text: of

text: the

text: Hilbert

text: space

text: H

text: N.

text: In

text: particular,

text: we

text: ﬁnd

text: that

text: p

text: q

text: Ker

text: C

text: iλ,

text: β)

text: q

text: q

text: (q,

text: λ)

text: and

text: β

text: (q,

text: Rq)

text: β

text: (q,

text: Rq)

text: (q,

text: ηq)

text: λ

text: (q,

text: αq)

text: (q,

text: αq)

text: (q,

text: αq)

text: Thus

text: we

text: ﬁnd

text: that

text: a

text: necessary

text: and

text: suﬃcient

text: condition

text: for

text: an

text: eigenvalue

text: ζ

text: of

text: the

text: system

text: operator

text: A

text: (β)

text: to

text: satisfy

text: Re

text: ζ

text: is

text: β

text: (q,

text: Rq)

text: (q,

text: ηq)

text: for

text: some

text: q

text: Ker

text: C

text: (ζ,

text: β)

text: q

text: (q,

text: αq)

text: (q,

text: αq)

text: Now

text: there

text: are

text: some

text: fundamental

text: inequalities

text: that

text: play

text: a

text: key

text: role

text: in

text: our

text: over-

text: damping

text: analysis

text: which

text: are

text: described

text: in

text: the

text: following

text: proposition.

text: We

text: will

text: prove

text: part

text: of

text: this

text: proposition,

text: the

text: rest

text: is

text: proved

text: in

text: Section

text: since

text: the

text: proof

text: is

text: enlightening

text: showing

text: how

text: the

text: energy

text: conservation

text: law

text: and

text: the

text: virial

text: theorem

text: can

text: be

text: used

text: to

text: derive

text: these

text: inequalities.

text: Proposition

text: (fundamental

text: inequalities)

text: The

text: following

text: statements

text: are

text: true.

text: C

text: For

text: any

text: q

text: N

text: with

text: q

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Overview of main results. Before we proceed, let us now summarize the results in this section. First, its obvious from this discussion that a necessary condition for (cid:29) the low-loss eigenmode v (t, β) to be overdamped for β 1 is ρ = 0 (even though it j − j does so with a slow damping of order β 1, as seen in (140) and (99)), but what is not − obvious is that this is in fact a suﬃcient condition. Although in the case η 1 exists { } (i.e., Ker η = 0 ), it just follows from the fourth statement in Proposition 16 since if → | | (cid:29) ρ = 0 then by (140) we have ζ (β) 0 which implies ζ (β) < ω for β 1 and j (cid:29) j j mi(cid:54) n { } hence Re ζ (β) = 0 for β 1. But this argument fails when Ker η = 0 . Thus we j must ﬁnd an alternative method to show that ρ = 0 is equivalent to Re ζ (β) = 0 (cid:29) j j for β 1. The key idea that leads to an alternative method is to realize that by − the virial theorem 1 and the ﬁfth statement in Proposition 16, when η 1 exists the modes with ρ = 0 are exactly the modes that break the energy equipartition. j { } Hence our general method is to show, even in the case Ker η = 0 , that for a low-loss modes with ρ = 0, there is a breakdown of the virial theorem 1 since as j → ∞ losses increase, i.e., as β , the equality between the kinetic and potential energy (see Corollary 2) of the corresponding eigenmode of the Lagrangian system (19) can no longer be maintained and must eventually break which forces the modes to be (cid:29) non-oscillatory for β 1, i.e., overdamped. This proves that ρ = 0 is equivalent to (cid:29) j Re ζ (β) = 0 for β 1. j Thus it follows that the low-loss eigenmodes in (105) of the canonical system (33) (cid:29) which are overdamped for β 1, their limiting vectors (satisfy lim →∞ v (t, β) = ∈ β j w˚ Ker Ω for t ﬁxed) form a orthonormal basis for Ker Ω . This implies of course j 1 1 that κ = dim Ker Ω is the number of low-loss eigenmodes that are overdamped for (cid:29) 1 β 1. A major result of this section is that in fact κ = N , where N = rank B = R R rank R. But N is also the number of high-loss eigenpairs in (96). Thus we conclude R using these results and Theorem 20 that the number of eigenmodes in (105) which (cid:29) are overdamped in the high-loss regime β 1 is exactly 2N with N of these being R R low-loss eigenmodes and the rest being all the high-loss eigenmodes. Moreover, in − the case N < N, the remaining 2N 2N > 0 modes are low-loss oscillatory modes R R with an extremely high quality factor that actually increases as the losses increase, → ∞ i.e., as β . Main results. We now give the statements of our main results. The proof of the ﬁrst theorem we give in this section as it is enlightening. The proofs of the rest of the statements below are found in Section 7. Theorem 22 (overdamped low-loss modes) A necessary and suﬃcient condi- − tion for a low-loss eigenmode v (t, β) = w (β) e iζj(β)t from (105) to be overdamped (cid:29) j ∈ j for β 1 is ρ = 0 (or equivalently, w˚ Ker Ω ). j j 1 − Proof. Let v (t, β) = w (β) e iζj(β)t be a low-loss eigenmode from (105). If it is j (cid:29) j (cid:29) overdamped for β 1 then Re ζ (β) = 0 for β 1 (i.e., for all β suﬃciently large) j ∈ and so by (140), (141) we have ρ = 0 and w˚ Ker Ω (the nullspace of Ω ). j j 1 1 37

text: Overview

text: of

text: main

text: results.

text: Before

text: we

text: proceed,

text: let

text: us

text: now

text: summarize

text: the

text: results

text: in

text: this

text: section.

text: First,

text: its

text: obvious

text: from

text: this

text: discussion

text: that

text: a

text: necessary

text: condition

text: for

text: the

text: low-loss

text: eigenmode

text: v

text: (t,

text: β)

text: to

text: be

text: overdamped

text: for

text: β

text: is

text: ρ

text: (even

text: though

text: it

text: j

text: j

text: does

text: so

text: with

text: a

text: slow

text: damping

text: of

text: order

text: β

text: as

text: seen

text: in

text: and

text: but

text: what

text: is

text: not

text: obvious

text: is

text: that

text: this

text: is

text: in

text: fact

text: a

text: suﬃcient

text: condition.

text: Although

text: in

text: the

text: case

text: η

text: exists

text: (i.e.,

text: Ker

text: η

text: it

text: just

text: follows

text: from

text: the

text: fourth

text: statement

text: in

text: Proposition

text: since

text: if

text: ρ

text: then

text: by

text: we

text: have

text: ζ

text: (β)

text: which

text: implies

text: ζ

text: (β)

text: ω

text: for

text: β

text: and

text: j

text: j

text: j

text: n

text: hence

text: Re

text: ζ

text: (β)

text: for

text: β

text: But

text: this

text: argument

text: fails

text: when

text: Ker

text: η

text: Thus

text: we

text: j

text: must

text: ﬁnd

text: an

text: alternative

text: method

text: to

text: show

text: that

text: ρ

text: is

text: equivalent

text: to

text: Re

text: ζ

text: (β)

text: j

text: j

text: for

text: β

text: The

text: key

text: idea

text: that

text: leads

text: to

text: an

text: alternative

text: method

text: is

text: to

text: realize

text: that

text: by

text: the

text: virial

text: theorem

text: and

text: the

text: ﬁfth

text: statement

text: in

text: Proposition

text: when

text: η

text: exists

text: the

text: modes

text: with

text: ρ

text: are

text: exactly

text: the

text: modes

text: that

text: break

text: the

text: energy

text: equipartition.

text: j

text: Hence

text: our

text: general

text: method

text: is

text: to

text: show,

text: even

text: in

text: the

text: case

text: Ker

text: η

text: that

text: for

text: a

text: low-loss

text: modes

text: with

text: ρ

text: there

text: is

text: a

text: breakdown

text: of

text: the

text: virial

text: theorem

text: since

text: as

text: j

text: losses

text: increase,

text: i.e.,

text: as

text: β

text: the

text: equality

text: between

text: the

text: kinetic

text: and

text: potential

text: energy

text: (see

text: Corollary

text: of

text: the

text: corresponding

text: eigenmode

text: of

text: the

text: Lagrangian

text: system

text: can

text: no

text: longer

text: be

text: maintained

text: and

text: must

text: eventually

text: break

text: which

text: forces

text: the

text: modes

text: to

text: be

text: non-oscillatory

text: for

text: β

text: i.e.,

text: overdamped.

text: This

text: proves

text: that

text: ρ

text: is

text: equivalent

text: to

text: j

text: Re

text: ζ

text: (β)

text: for

text: β

text: j

text: Thus

text: it

text: follows

text: that

text: the

text: low-loss

text: eigenmodes

text: in

text: of

text: the

text: canonical

text: system

text: which

text: are

text: overdamped

text: for

text: β

text: their

text: limiting

text: vectors

text: (satisfy

text: lim

text: v

text: (t,

text: β)

text: β

text: j

text: w˚

text: Ker

text: Ω

text: for

text: t

text: ﬁxed)

text: form

text: a

text: orthonormal

text: basis

text: for

text: Ker

text: Ω

text: This

text: implies

text: of

text: course

text: j

text: that

text: κ

text: dim

text: Ker

text: Ω

text: is

text: the

text: number

text: of

text: low-loss

text: eigenmodes

text: that

text: are

text: overdamped

text: for

text: β

text: A

text: major

text: result

text: of

text: this

text: section

text: is

text: that

text: in

text: fact

text: κ

text: N

text: where

text: N

text: rank

text: B

text: R

text: R

text: rank

text: R.

text: But

text: N

text: is

text: also

text: the

text: number

text: of

text: high-loss

text: eigenpairs

text: in

text: Thus

text: we

text: conclude

text: R

text: using

text: these

text: results

text: and

text: Theorem

text: that

text: the

text: number

text: of

text: eigenmodes

text: in

text: which

text: are

text: overdamped

text: in

text: the

text: high-loss

text: regime

text: β

text: is

text: exactly

text: with

text: N

text: of

text: these

text: being

text: R

text: R

text: low-loss

text: eigenmodes

text: and

text: the

text: rest

text: being

text: all

text: the

text: high-loss

text: eigenmodes.

text: Moreover,

text: in

text: the

text: case

text: N

text: N,

text: the

text: remaining

text: modes

text: are

text: low-loss

text: oscillatory

text: modes

text: R

text: R

text: with

text: an

text: extremely

text: high

text: quality

text: factor

text: that

text: actually

text: increases

text: as

text: the

text: losses

text: increase,

text: i.e.,

text: as

text: β

text: Main

text: results.

text: We

text: now

text: give

text: the

text: statements

text: of

text: our

text: main

text: results.

text: The

text: proof

text: of

text: the

text: ﬁrst

text: theorem

text: we

text: give

text: in

text: this

text: section

text: as

text: it

text: is

text: enlightening.

text: The

text: proofs

text: of

text: the

text: rest

text: of

text: the

text: statements

text: below

text: are

text: found

text: in

text: Section

text: Theorem

text: (overdamped

text: low-loss

text: modes)

text: A

text: necessary

text: and

text: suﬃcient

text: condi-

text: tion

text: for

text: a

text: low-loss

text: eigenmode

text: v

text: (t,

text: β)

text: w

text: (β)

text: e

text: iζj(β)t

text: from

text: to

text: be

text: overdamped

text: j

text: j

text: for

text: β

text: is

text: ρ

text: (or

text: equivalently,

text: w˚

text: Ker

text: Ω

text: j

text: j

text: Proof.

text: Let

text: v

text: (t,

text: β)

text: w

text: (β)

text: e

text: iζj(β)t

text: be

text: a

text: low-loss

text: eigenmode

text: from

text: If

text: it

text: is

text: j

text: j

text: overdamped

text: for

text: β

text: then

text: Re

text: ζ

text: (β)

text: for

text: β

text: (i.e.,

text: for

text: all

text: β

text: suﬃciently

text: large)

text: j

text: and

text: so

text: by

text: we

text: have

text: ρ

text: and

text: w˚

text: Ker

text: Ω

text: (the

text: nullspace

text: of

text: Ω

text: j

text: j


text: ⊕ ⊥ ⊥ ⊥ H = H H that P and P ΩP with respect to this decomposition are the block p q B B B operator(cid:20)s (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) ⊥ ⊥ − ⊥ − ⊥ ⊥ P 0 ⊥ ⊥ P 0 0 iΦT P 0 0 iP ΦT P = R˜ , P ΩP = R˜ R˜ = ⊥ R˜ . B 0 1 B B 0 1 iΦ 0 0 1 iΦP 0 R˜ (170) (cid:26)(cid:20) (cid:21) (cid:27) From (169), (170) it follows that ϕ ∈ ⊕ − ⊥ ⊥ Ker Ω = Ker R˜ H : iP ΦTψ = 0 and iΦP ϕ = 0 . 1 ψ q R˜ R˜ ⊥ ∈ ⊥ ⊥ But if iΦP ϕ = 0 with ϕ Ker R˜ then P ϕ = ϕ by deﬁnition of P and hence R˜ R˜ R˜ R˜ϕ = 0 and Φϕ = 0 which implies ηKTϕ =√0 and RKTϕ = 0√since by (35) and (30) p − p we have Φ = K KT, R˜ = K RKT, K = α 1, and K = η. By the hypothesis q p p p p q of Condition 21 this implies that ϕ = 0. Also, since R˜ is a Hermitian matrix it − ⊥ ⊥ follows that I P is the orthogonal projection onto th√e Ran R˜ so that P ΦTψ = 0 R˜ ∈ ∈ R˜ is equivalen√t to ΦTψ Ran R˜ and this is√equivalent to ηψ √Ran R by properties ∈ ∈ ∩ (31). But η√ψ Ran R is equivalent to ηψ Ran R Ran η which by (162) is ∈ ∩ equivalent to ηψ Ran R Ran η. This proves (166). Finally, we will now prove (168). It follows from th√e fact that η is a H√ermitian | matrix and (162) that the restriction of the operator η to Ran η, i.e., η : → Ranη Ran η Ran η, is an invertible operator and that √ √ { ∈ ∈ ∩ } |− ∩ φ Ran η : ηφ Ran η Ran R = η 1 (Ran η Ran R) . (171) Ranη From this it follows that √ { ∈ ∈ ∩ } ∩ dim φ Ran η : ηφ Ran η Ran R = dim (Ran η Ran R) . The proof of (168) immediately follows from this fact and (165). This completes the proof of the theorem. 8 Appendix: Schur complement and the Aitken formula (cid:20) (cid:21) Let P Q M = (172) R S 46 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: H

text: H

text: H

text: that

text: P

text: and

text: P

text: ΩP

text: with

text: respect

text: to

text: this

text: decomposition

text: are

text: the

text: block

text: p

text: q

text: B

text: B

text: B

text: P

text: P

text: iΦT

text: P

text: iP

text: ΦT

text: P

text: R˜

text: P

text: ΩP

text: R˜

text: R˜

text: R˜

text: B

text: B

text: B

text: iΦ

text: iΦP

text: R˜

text: From

text: it

text: follows

text: that

text: ϕ

text: Ker

text: Ω

text: Ker

text: R˜

text: H

text: iP

text: ΦTψ

text: and

text: iΦP

text: ϕ

text: ψ

text: q

text: R˜

text: R˜

text: But

text: if

text: iΦP

text: ϕ

text: with

text: ϕ

text: Ker

text: R˜

text: then

text: P

text: ϕ

text: ϕ

text: by

text: deﬁnition

text: of

text: P

text: and

text: hence

text: R˜

text: R˜

text: R˜

text: R˜ϕ

text: and

text: Φϕ

text: which

text: implies

text: ηKTϕ

text: and

text: RKTϕ

text: by

text: and

text: p

text: p

text: we

text: have

text: Φ

text: K

text: KT,

text: R˜

text: K

text: RKT,

text: K

text: α

text: and

text: K

text: η.

text: By

text: the

text: hypothesis

text: q

text: p

text: p

text: p

text: p

text: q

text: of

text: Condition

text: this

text: implies

text: that

text: ϕ

text: Also,

text: since

text: R˜

text: is

text: a

text: Hermitian

text: matrix

text: it

text: follows

text: that

text: I

text: P

text: is

text: the

text: orthogonal

text: projection

text: onto

text: th√e

text: Ran

text: R˜

text: so

text: that

text: P

text: ΦTψ

text: R˜

text: R˜

text: is

text: equivalen√t

text: to

text: ΦTψ

text: Ran

text: R˜

text: and

text: this

text: is√equivalent

text: to

text: ηψ

text: √Ran

text: R

text: by

text: properties

text: But

text: η√ψ

text: Ran

text: R

text: is

text: equivalent

text: to

text: ηψ

text: Ran

text: R

text: Ran

text: η

text: which

text: by

text: is

text: equivalent

text: to

text: ηψ

text: Ran

text: R

text: Ran

text: η.

text: This

text: proves

text: Finally,

text: we

text: will

text: now

text: prove

text: It

text: follows

text: from

text: th√e

text: fact

text: that

text: η

text: is

text: a

text: H√ermitian

text: matrix

text: and

text: that

text: the

text: restriction

text: of

text: the

text: operator

text: η

text: to

text: Ran

text: η,

text: i.e.,

text: η

text: Ranη

text: Ran

text: η

text: Ran

text: η,

text: is

text: an

text: invertible

text: operator

text: and

text: that

text: φ

text: Ran

text: η

text: ηφ

text: Ran

text: η

text: Ran

text: R

text: η

text: (Ran

text: η

text: Ran

text: R)

text: Ranη

text: From

text: this

text: it

text: follows

text: that

text: dim

text: φ

text: Ran

text: η

text: ηφ

text: Ran

text: η

text: Ran

text: R

text: dim

text: (Ran

text: η

text: Ran

text: R)

text: The

text: proof

text: of

text: immediately

text: follows

text: from

text: this

text: fact

text: and

text: This

text: completes

text: the

text: proof

text: of

text: the

text: theorem.

text: Appendix:

text: Schur

text: complement

text: and

text: the

text: Aitken

text: formula

text: Let

text: P

text: Q

text: M

text: R

text: S

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Then we arrive at the following uniquely deﬁned expressions for the kinetic and potential energies L L T 1 ∂ V 1 ∂ − L ˙ ˙ = Q, = Q . (195) 2 ∂Q˙ 2 ∂Q˙ (cid:16) (cid:17) We are particularly interested in the case of the quadratic Lagrangian L L 1 − 1 = Q, Q˙ = Q˙ TαQ˙ + Q˙ TθQ QTηQ, (196) 2 2 where α and η are real symmetric matrices which are positive deﬁnite and positive semideﬁnite, respectively, and θ is a real skew-symmetric matrix, that is ≥ − αT = α > 0, ηT = η 0, θT = θ, (197) H Then according to (194) the total energy the takes the form H 1 1 ≥ = Q˙ TαQ˙ + QTηQ 0. (198) 2 2 Consequently, as it follows from (193) the kinetic and the potential energies are deﬁned by T 1 1 V 1 − 1 = Q˙ TαQ˙ + Q˙ TθQ, = QTηQ Q˙ TθQ. (199) 2 2 2 2 The Euler-Lagrange equations corresponding to the quadratic Lagrangian (196) are ¨ ˙ αQ + 2θQ + ηQ = 0. (200) Multiplying the above equation by QT from the left yields QTαQ¨ + 2QTθQ˙ + QTηQ = 0, (201) (cid:16) (cid:17) and this equation can be recast recast as − − ∂ QTαQ˙ Q˙ TαQ˙ 2Q˙ TθQ + QTηQ = 0. (202) t Combining the above equation with (193) and (196) we arrive at the following important identity that holds for any solution Q to the Euler-Lagrange equation (200) 56 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Then

text: we

text: arrive

text: at

text: the

text: following

text: uniquely

text: deﬁned

text: expressions

text: for

text: the

text: kinetic

text: and

text: potential

text: energies

text: L

text: L

text: T

text: V

text: L

text: Q,

text: Q

text: ∂Q˙

text: ∂Q˙

text: We

text: are

text: particularly

text: interested

text: in

text: the

text: case

text: of

text: the

text: quadratic

text: Lagrangian

text: L

text: L

text: Q,

text: Q˙

text: Q˙

text: TαQ˙

text: Q˙

text: TθQ

text: QTηQ,

text: where

text: α

text: and

text: η

text: are

text: real

text: symmetric

text: matrices

text: which

text: are

text: positive

text: deﬁnite

text: and

text: positive

text: semideﬁnite,

text: respectively,

text: and

text: θ

text: is

text: a

text: real

text: skew-symmetric

text: matrix,

text: that

text: is

text: αT

text: α

text: ηT

text: η

text: θT

text: θ,

text: H

text: Then

text: according

text: to

text: the

text: total

text: energy

text: the

text: takes

text: the

text: form

text: H

text: Q˙

text: TαQ˙

text: QTηQ

text: Consequently,

text: as

text: it

text: follows

text: from

text: the

text: kinetic

text: and

text: the

text: potential

text: energies

text: are

text: deﬁned

text: by

text: T

text: V

text: Q˙

text: TαQ˙

text: Q˙

text: TθQ,

text: QTηQ

text: Q˙

text: TθQ.

text: The

text: Euler-Lagrange

text: equations

text: corresponding

text: to

text: the

text: quadratic

text: Lagrangian

text: are

text: αQ

text: ηQ

text: Multiplying

text: the

text: above

text: equation

text: by

text: QT

text: from

text: the

text: left

text: yields

text: QTαQ¨

text: QTηQ

text: and

text: this

text: equation

text: can

text: be

text: recast

text: recast

text: as

text: QTαQ˙

text: Q˙

text: TαQ˙

text: TθQ

text: QTηQ

text: t

text: Combining

text: the

text: above

text: equation

text: with

text: and

text: we

text: arrive

text: at

text: the

text: following

text: important

text: identity

text: that

text: holds

text: for

text: any

text: solution

text: Q

text: to

text: the

text: Euler-Lagrange

text: equation

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [Bau85] H. Baumgartel, Analytic Perturbation Theory for Matrices and Oper- ators, Birkh¨auser Verlag, Basel 1985. [Duﬀ55] R. J. Duﬃn, A minimax theory for overdamped networks, J. Rational Mech. Anal. 4, 221–233, (1955). [FigSch2] A. Figotin and J. H. Schenker, Hamiltonian structure for dispersive and dissipative dynamical systems, J. Stat. Phys., 128, 969-1056, (2007). [FigVit1] A. Figotin and I. Vitebskiy, Spectra of Periodic Nonreciprocal Electric Circuits, SIAM, 61, 6, 2008–2035, (2001). [FigVit8] A. Figotin and I. Vitebskiy, Absorption suppression in photonic crys- tals, Phys. Rev. B, 77, 104421 (2008). [Gant] F. Gantmacher, Lectures in Analytical Mechanics, Mir, 1975. [Gold] H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd Ed., Addison-Wesley, 2001. [FigWel1] A. Figotin and A. Welters, Dissipative properties of systems com- posed of high-loss and lossless components, J. Math. Phys. 53, 123508 (2012). [Kato] T. Kato, Perturbation theory for linear operators, reprint of the 1980 edition, classics in mathematics, Springer-Verlag, Berlin, 1995. [Lan66] P. Lancaster, Lambda-matrices and vibrating systems, Dover Publi- cations, Inc., Mineola, NY, 2002. [Mar88] A. S. Markus, Introduction to the spectral theory of polynomial oper- ator pencils, Translations of Mathematical Monographs, 71, American Mathematical Society, Providence, RI, 1988. [Pain] J. Pain, The Physics of Vibrations and Waves, 6th Ed., Wiley, 2004. [Pars] L. Pars, Treatise on Analytical Mechanics, Heinemann, 1965. [Poincare FS] H. Poincar´e, The Foundations of Science, authorized translation to English by B. Halsted, The Science Press, 1913. [ReSi1] M. Reed and B. Simon, Methods of modern mathematical physics, Vol. 1: Functional analysis, Second edition, Academic Press, Inc., New York, 1980. 58

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