Compression Theorem Research Articles

The compression theorem I

This the first of a set of three papers about the Compression Theorem: if M^m is embedded in Q^q X R with a normal vector field and if q-m > 0, then the given vector field can be straightened (ie, made parallel to the given R direction) by an isotopy of M and normal field in Q X R. The theorem can be deduced from Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer-Verlag (1986); 2.4.5 C'] and is implicit in the preceeding discussion. Here we give a direct proof that leads to an explicit description of the finishing embedding. In the second paper in the series we give a proof in the spirit of Gromov's proof and in the third part we give applications.

The compression theorem II: directed embeddings

This is the second of three papers about the Compression Theorem. We give proofs of Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer--Verlag (1986); 2.4.5 C'] and of the Normal Deformation Theorem [The compression theorem I; 4.7], arxiv:math.GT/9712235.

Equivariant Configuration Spaces

The compression theorem is used to prove results for equivariant configuration spaces that are analogous to the well-known non-equivariant results of May, Milgram and Segal.

Loop spaces and the compression theorem

For a smooth, finite-dimensional manifold M M with a submanifold S S we study the topology of the straight loop space Ω S s t M \Omega ^{st}_SM , the space of loops whose intersections with S S are subject to a certain transversality condition. Our main tool is Rourke and Sanderson’s compression theorem. We prove that the homotopy type of the straight loop space of a link in S 3 S^3 depends only on the number of link components.

A note on the compression theorem for convex surfaces

Suppose a ib ic i (i=1,2) are two triangles of equal side lengths and lying on spheres Φ i with radii r 1,r 2 (r 1<r 2) , respectively. We have proved that there is a continuous map h of a 1 b 1 c 1 onto a 2 b 2 c 2 so that for any two points p, q in a 1b 1c 1, |pq|⩾|h(p)h(q)| (Rubinstein and Weng, J. Combin. Optim. 1 (1997) 67–78). In this note we generalize this compression theorem to convex surfaces.

Essential and inessential maps and continuation theorems

The notions of essential and inessential maps are given for new classes of maps. In addition, generalizations of Krasnoselskii's compression and expansion theorem in a cone are presented.

Resolving vibrational and structural contributions to isothermal compressibility

The well-known and general “compressibility theorem” for pure substances relates κT =−(∂ ln V/∂p)N,T to a spatial integral involving the pair correlation function g(2). The isochoric inherent structure formalism for condensed phases separates g(2) into two fundamentally distinct contributions: a generally anharmonic vibrational part, and a structural relaxation part. Only the former determines κT for low-temperature crystals, but both operate in the liquid phase. As a supercooled liquid passes downward in temperature through a glass transition, the structural contribution to κT switches off to produce the experimentally familiar drop in this quantity. The Kirkwood–Buff solution theory forms the starting point for extension to mixtures, with electroneutrality conditions creating simplifications in the case of ionic systems.

A rate-distortion theorem for arbitrary discrete sources

A rate-distortion theorem for arbitrary (not necessarily stationary or ergodic) discrete-time finite-alphabet sources is given. This result, which provides the expression of the minimum /spl epsiv/-achievable fixed-length coding rate subject to a fidelity criterion, extends a recent data compression theorem by Steinberg and Verdu (see ibid., vol.42, p.63-86 (Jan. 1996).

Generalized source coding theorems and hypothesis testing: Part II — Operational limits

In light of the information measures introduced in Part I, a generalized version of the Asymptotic Equipartition Property (AEP) is proved. General fixed‐length data compaction and data compression (source coding) theorems for arbitrary finite‐alphabet sources are also established. Finally, the general expression of the Neyman‐Pearson type‐II error exponent subject to upper bounds on the type‐I error probability is examined.

Compression Theorems and Steiner Ratios on Spheres

Suppose AiBiCi (i = 1, 2) are two triangles of equal side lengths lying on spheres Φi with radii r1, r2 (r1 < r2) respectively. First we prove the existence of a map h: A1B1C1 → A2B2C2 so that for any two points P1, Q1 in A1B1C1,¦P1Q1¦≥¦h(P1)h(Q1)¦. Moreover, if P1, Q1 are not on the same side, then the inequality strictly holds. This compression theorem can be applied to compare the minimum of a variable in triangles on two spheres. Hence, one of the applications of the compression theorem is the study of Steiner minimal tress on spheres. The Steiner ratio is the largest lower bound for the ratio of the lengths of Steiner minimal trees to minimal spanning trees for point sets in a metric space. Using the compression theorem we prove that the Steiner ratio on spheres is the same as on the Euclidean plane, namely \(\backslash \bar 3/2\).

Effective category and measure in abstract complexity theory

Strong variants of the Operator Speed-up Theorem, Operator Gap Theorem and Compression Theorem are obtained using an effective version of Baire Category Theorem. It is also shown that all complexity classes of recursive predicates have effective measure zero in the space of recursive predicates and, on the other hand, the class of predicates with almost everywhere complexity above an arbitrary recursive threshold has recursive measure one in the class of recursive predicates.

On the determination of effective hard convex body parameters for liquid mixtures

We derive a systematic method for determining effective molecular size parameters for use in perturbation theories of dense fluid mixtures using non-spherical reference systems of hard convex bodies. The method is based on the compressibility theorem. For the case of mixtures of atomic fluids, the method reduces to a prescription suggested earlier by one of the authors. For molecular fluid mixtures where a reference system of hard ellipsoids is employed, a sufficient set of equations to determine the size and shape of the equivalent ellipsoids is derived. The method is applied to the benchmark system of an equimolar mixture of Ar/Kr at 115·8 K.

On the properties of the bridge function in simple fluids

Using functional variation methods an exact functional equation has been derived for the bridge function in an inhomogeneous fluid. Step by step the equation has been simplified using the virial theorem as a closure relation. As a result a closed but approximate equation has been given that expresses a fully consistent bridge function in terms of the correlation function and its derivatives. In the homogeneous limit new and exact global relations have been derived by requiring consistency with respect to the energy, the virial and the compressibility theorem.

Totally asynchronous Slepian-Wolf data compression

It is proved that the Slepian-Wolf data compression theorem still holds when both encoders are operating totally asynchronously. In addition it is shown that in this case the Wyner-Ahlswede-Korner source-coding theorem holds. >

The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem

A Cohen-Macaulay complex is said to be balanced of typea=(a 1,a 2, ...,a s ) if its vertices can be colored usings colors so that every maximal face gets exactlya i vertices of thei:th color. Forb=(b 1,b 2, ...,b s ), 0≦b≦a, letf b denote the number of faces havingb i vertices of thei:th color. Our main result gives a characterization of thef-vectorsf=(f b )0≦b≦a′ or equivalently theh-vectors, which can arise in this way from balanced Cohen-Macaulay complexes. As part of the proof we establish a generalization of Macaulay’s compression theorem to colored multicomplexes. Finally, a combinatorial shifting technique for multicomplexes is used to give a new simple proof of the original Macaulay theorem and another closely related result.

Remarks on the hypernetted chain approximation in the case of simple fluids

In the theory of simple fluids it is a common feature of approximate theories that the energy, the virial and the compressibility theorem give not the same equation of state as it should be. It is shown that in general two conditions must be satisfied in order to remove the inconsistency. Furthermore it is noted that Morita's and Hiroïke's expression for the Helmholtz free energy should be corrected in the case of hard sphere repulsive forces. This turns out to have consequences for the H.N.C. approximation. Lastly, the variational formulation of theory is discussed in view of consistency.

Partial and Total Matrix Multiplication

In 1979 considerable progress was made in estimating the complexity of matrix multiplication. Here the new techniques and recent results are presented, based upon the notion of approximate rank and the observation that certain patterns of partial matrix multiplication (some of the entries of the matrices may be zero) can efficiently be utilized to perform multiplication of large total matrices. By combining Pan’s trilinear technique with a strong version of our compression theorem for the case of several disjoint matrix multiplications it is shown that multiplication of $N \times N$ matrices (over arbitrary fields) is possible in time $O(N^\beta )$, where $\beta $ is a bit smaller than $3\ln 52/\ln 110 \approx 2.522$.

Nonexistence of program optimizers in several abstract settings

For various settings and various dynamic criteria for gauging optimality of programs,there does not exist a master program (optimizer) such that if P is any program which computes a partial function pocessing an optimal program, then , operating on the program P as input, halts eventually and outputs an optimal program P for computing that partial function. Optimality can be gauged by a criterion suggested by a variant of M. Blum's compression theorem for an arbitrary complexity measure, by optimality except for a linear factor for amount of memory used by a Turing machine, or by optimality within ε on a RASP. Thus, our techniques are compatible with techniques for producing optimal programs which are as diverse as upward diagonalization, downward diagonalization, and the size arguments of Hartmanis. Our nonexistence results continue to hold even if we only ask that an optimizer behave properly when the input program P satisfies certain convergence properties (e.g., when P computes a total function) and possesses an equivalent optimal program which is neither too hard nor too easy to compute.

Relativization of the theory of computational complexity

Blum''s machine-independent treatment of the complexity of partial recursive functions is extended to relative algorithms (as represented by Turing machines with oracles). We prove relativizations of several results of Blum complexity theory, such as the compression theorem. A recursive relatedness theorem is proved, showing that any two relative complexity measures are related by a fixed recursive function. This theorem allows us to obtain proofs of results for all measures from proofs for a particular measure. We study complexity-determined reducibilities, the parallel notion to complexity classes for the relativized case. Truth-table and primitive recursive reducibilities are reducibilities of this type, while other commonly-studied reducibilities are not. We formalize the concept of a set the computation of function (by causing a saving in resource when used as an oracle in the computation of the function). Basic properties of the helping relation are proved, including non-transitivity and bounds on the amount of help certain sets can provide. Several independence results (results about sets that don''t help each other''s computation) are proved; they are subrecursive analogs to degrees-of-unsolvability theorems, with similar proofs using diagonalization and priority arguments. In particular, we discuss the existence of a universally-helped set, obtaining partial results in both directions. The deepest result is a finite-injury priority argument (without an apparent recursive bound on the number of injuries) which produces sets preserving an arbitrary lower bound on the complexity of a set. Our methods of proof include proof for a simple measure (e.g. space) and appeal to recursive relatedness, diagonalization and priority techniques, and heavy use of arguments about the domain of convergence of partial recursive functions in order to define total recursive functions.

A proof of the data compression theorem of Slepian and Wolf for ergodic sources (Corresp.)

If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{(X_i, Y_i)\}_{i=1}^{\infty}</tex> is a sequence of independent identically distributed discrete random pairs with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(X_i, Y_i) ~ p(x,y)</tex> , Slepian and Wolf have shown that the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> process and the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</tex> process can be separately described to a common receiver at rates <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_X</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_Y</tex> hits per symbol if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_X + R_Y &gt; H(X,Y), R_X &gt; H(X\midY), R_Y &gt; H(Y\midX)</tex> . A simpler proof of this result will be given. As a consequence it is established that the Slepian-Wolf theorem is true without change for arbitrary ergodic processes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{(X_i,Y_i)\}_{i=1}^{\infty}</tex> and countably infinite alphabets. The extension to an arbitrary number of processes is immediate.