In 1975, Richard O'Malley proved that every approximately continuous function has approximate extrema, and this result provides an immediate solution to Scottish Book Problem 157. The purpose of this paper is to provide an additional proof of O'Malley's result.

In 1971, D. Ornstein proved a theorem that completely solved Problem 157 of the Scottish Book. The purpose of this paper is to give an independent proof.

Determining the achievable rate region for networks using routing, linear coding, or nonlinear coding is thought to be a difficult task in general, and few are known. We describe the achievable rate regions for four interesting networks (completely for three and partially for the fourth). In addition to the known matrix-computation method for proving outer bounds for linear coding, we present a new method that yields actual characteristic-dependent linear rank inequalities from which the desired bounds follow immediately.

Casinos operate by generating sequences of outcomes which appear unpredictable, or random, to effective gamblers. We investigate relative notions of randomness for gamblers whose wagers are restricted to a finite set. Some sequences which appear unpredictable to gamblers using wager amounts in one set permit unbounded profits for gamblers using different wager values. In particular, we show that for non-empty finite sets A and B, every A-valued random is B-valued random if and only if there exists a k⩾0 such that B⊆A⋅k.

Networks derived from matroids have played a fundamental role in proving theoretical results about the limits of network coding. In this tutorial paper, we review many connections between matroids and network coding theory, with specific emphasis on network solvability, admissible network alphabet sizes, linear coding, and network capacity.

2 Matroid Theory 3 This paper explores the connection between network coding and Matroid theory, 4 a branch of mathematics that generalizes linear algebra and graph theory. ABSTRACT | Networks derived from matroids have played a 7 fundamental role in proving theoretical results about the limits 8 of network coding. In this tutorial paper, we review many con-9 nections between matroids and network coding theory, with 10 specific emphasis on network solvability, admissible network 11 alphabet sizes, linear coding, and network capacity.

Any unconstrained information inequality in three or fewer random variables can be written as a linear combination of instances of Shannon's inequality I(A;B|C) >= 0 . Such inequalities are sometimes referred to as "Shannon" inequalities. In 1998, Zhang and Yeung gave the first example of a "non-Shannon" information inequality in four variables. Their technique was to add two auxiliary variables with special properties and then apply Shannon inequalities to the enlarged list. Here we will show that the Zhang-Yeung inequality can actually be derived from just one auxiliary variable. Then we use their same basic technique of adding auxiliary variables to give many other non-Shannon inequalities in four variables. Our list includes the inequalities found by Xu, Wang, and Sun, but it is by no means exhaustive. Furthermore, some of the inequalities obtained may be superseded by stronger inequalities that have yet to be found. Indeed, we show that the Zhang-Yeung inequality is one of those that is superseded. We also present several infinite families of inequalities. This list includes some, but not all of the infinite families found by Matus. Then we will give a description of what additional information these inequalities tell us about entropy space. This will include a conjecture on the maximum possible failure of Ingleton's inequality. Finally, we will present an application of non-Shannon inequalities to network coding. We will demonstrate how these inequalities are useful in finding bounds on the information that can flow through a particular network called the Vamos network.

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsH. FejzićHAJRUDIN FEJZIć grew up in Bosnia. He received his Ph.D. from Michigan State University and has taught at CSUSB since 1994. He enjoys doing home improvement projects, gardening, and visiting national parks.C. FreilingCHRIS FREILING received his Ph.D. from University of California, Los Angeles, in 1981 and is now professor of mathematics at CSUSB. He also does consulting work for the Center for Communications Research in San Diego, CA. He loves God, family, friends, surfing (in water), and ultimate frisbee.D. RinneDAN RINNE received his Ph.D. from University of California, Santa Barbara, in 1979 and has taught at CSUSB since 1982. He reads a few pages of P. G. Wodehouse every day.

Ranks of subspaces of vector spaces satisfy all linear inequalities satisfied by entropies (including the standard Shannon inequalities) and an additional inequality due to Ingleton. It is known that the Shannon and Ingleton inequalities generate all such linear rank inequalities on up to four variables, but it has been an open question whether additional inequalities hold for the case of five or more variables. Here we give a list of 24 inequalities which, together with the Shannon and Ingleton inequalities, generate all linear rank inequalities on five variables. We also give a partial list of linear rank inequalities on six variables and general results which produce such inequalities on an arbitrary number of variables; we prove that there are essentially new inequalities at each number of variables beyond four (a result also proved recently by Kinser).

Functional differences that lead to generalized Riemann derivatives were studied by Ash and Jones in (1987). They gave a partial answer as to when these differences satisfy an analog of the Mean Value Theorem. Here we give a complete classification.

If beta and gamma are nonnegative integers and F is a field, then a polynomial collection {p 1 , hellip ,P beta } sube Z[alpha 1,hellip , alpha gamma ] is said to be solvable over F if there exist omega 1hellip , omega gamma isin F such that for all i = 1, hellip , beta we have p i (omega 1hellip , omega gamma ) = 0. We say that a network and a polynomial collection are solvably equivalent if for each field F the network has a scalar-linear solution over F if and only if the polynomial collection is solvable over F. Koetter and Medard's work implies that for any directed acyclic network, there exists a solvably equivalent polynomial collection. We provide the converse result, namely, that for any polynomial collection there exists a solvably equivalent directed acyclic network. (Hence, the problems of network scalar-linear solvability and polynomial collection solvability have the same complexity.) The construction of the network is modeled on a matroid construction using finite projective planes, due to MacLane in 1936. A set psi of prime numbers is a set of characteristics of a network if for every q isin psi, the network has a scalar-linear solution over some finite field with characteristic q and does not have a scalar-linear solution over any finite field whose characteristic lies outside of psi. We show that a collection of primes is a set of characteristics of some network if and only if the collection is finite or co-finite. Two networks N and N' are Is-equivalent if for any finite field F, N is scalar-linearly solvable over F if and only if N' is scalar- linearly solvable over F. We further show that every network is ls-equivalent to a multiple-unicast matroidal network.

We define a class of networks, called matroidal networks, which includes as special cases all scalar-linearly solvable networks, and in particular solvable multicast networks. We then present a method for constructing matroidal networks from known matroids. We specifically construct networks that play an important role in proving results in the literature, such as the insufficiency of linear network coding and the unachievability of network coding capacity. We also construct a new network, from the Vamos matroid, which we call the Vamos network, and use it to prove that Shannon-type information inequalities are in general not sufficient for computing network coding capacities. To accomplish this, we obtain a capacity upper bound for the Vamos network using a non-Shannon-type information inequality discovered in 1998 by Zhang and Yeung, and then show that it is smaller than any such bound derived from Shannon-type information inequalities. This is the first application of a non-Shannon-type inequality to network coding. We also compute the exact routing capacity and linear coding capacity of the Vamos network. Finally, using a variation of the Vamos network, we prove that Shannon-type information inequalities are insufficient even for computing network coding capacities of multiple-unicast networks.

All unconstrained information inequalities in three or fewer random variables are known to be "Shannon-type", in that they are nonnegative linear combinations of instances of the inequality I(A;B|C) ges 0. In 1998, Zhang and Yeung gave the first example of an information inequality in four variables that is not "Shannon-type". Here we give six new unconstrained non-Shannon information inequalities in four variables. The new inequalities are independent of each other and of the Zhang-Yeung inequality

The coding capacity of a network is the supremum of ratios k/n, for which there exists a fractional (k,n) coding solution, where k is the source message dimension and n is the maximum edge dimension. The coding capacity is referred to as routing capacity in the case when only routing is allowed. A network is said to achieve its capacity if there is some fractional (k,n) solution for which k/n equals the capacity. The routing capacity is known to be achievable for arbitrary networks. We give an example of a network whose coding capacity (which is 1) cannot be achieved by a network code. We do this by constructing two networks, one of which is solvable if and only if the alphabet size is odd, and the other of which is solvable if and only if the alphabet size is a power of 2. No linearity assumptions are made.

The well-studied Vámos matroid has provided a wealth of interesting theoretical results in matroid theory. We use the Vámos matroid to construct a new network, which we call the Vámos network. We then exploit the Vámos network to answer in the negative the open question as to whether Shannon-type information inequalities are in general sufficient for computing network coding capacities. To accomplish this, we first determine the smallest coding capacity upper bound that can be obtained for the Váamos network using only Shannon-type information inequalities. Then, we prove that a smaller capacity upper bound for the Vámos network can be obtained by using a non-Shannon-type information inequality discovered in 1998 by Zhang and Yeung. This is the first published application of a non-Shannon-type inequality to network coding. Finally, we demonstrate that one can compute the exact routing capacity and linear coding capacity of the Vámos network.

The well-studied Vámos matroid has provided a wealth of interesting theoretical results in matroid theory. We use the Vámos matroid to construct a new network, which we call the Vámos network. We then exploit the Vámos network to answer in the negative the open question as to whether Shannon-type information inequalities are in general sufficient for computing network coding capacities. To accomplish this, we first determine the smallest coding capacity upper bound that can be obtained for the Váamos network using only Shannon-type information inequalities. Then, we prove that a smaller capacity upper bound for the Vámos network can be obtained by using a non-Shannon-type information inequality discovered in 1998 by Zhang and Yeung. This is the first published application of a non-Shannon-type inequality to network coding. Finally, we demonstrate that one can compute the exact routing capacity and linear coding capacity of the Vámos network.

It is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite field alphabet. It is also known that this result does not generalize to arbitrary networks. There are several examples in the literature of solvable networks with no scalar linear solution over any finite field. However, each example has a linear solution for some vector dimension greater than one. It has been conjectured that every solvable network has a linear solution over some finite field alphabet and some vector dimension. We provide a counterexample to this conjecture. We also show that if a network has no linear solution over any finite field, then it has no linear solution over any finite commutative ring with identity. Our counterexample network has no linear solution even in the more general algebraic context of modules, which includes as special cases all finite rings and Abelian groups. Furthermore, we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field (exactly 10% greater), so the network is not even asymptotically linearly solvable. It follows that, even for more general versions of linearity such as convolutional coding, filter-bank coding, or linear time sharing, the network has no linear solution