Steven M. Seubert

Multi-attribute mate choice decisions and uncertainty in the decision process: a generalized sequential search strategy

The behavior of females in search of a mate determines the likelihood that high quality males are encountered and adaptive search strategies rely on the effective use of available information on the quality of prospective mates. The sequential search strategy was formulated, like most models of search behavior, on the assumption that females obtain perfect information on the quality of encountered males. In this paper, we modify the strategy to allow for uncertainty of male quality and we determine how the magnitude of this uncertainty and the ability of females to inspect multiple male attributes to reduce uncertainty influence mate choice decisions. In general, searchers are sensitive to search costs and higher costs lower acceptance criteria under all versions of the model. The choosiness of searchers increases with the variability of the quality of prospective mates under conditions of the original model, but under conditions of uncertainty the choosiness of searchers may increase or decrease with the variability of inspected male attributes. The behavioral response depends on the functional relationship between observed male attributes and the fitness return to searchers and on costs associated with the search process. Higher uncertainty often induces searchers to pay more for information and under conditions of uncertainty the fitness return to searchers is never higher than under conditions of the original model. Further studies of the performance of alternative search strategies under conditions of uncertainty may consequently be necessary to identify search strategies likely to be used under natural conditions.

Mate choice and optimal search behavior: Fitness returns under the fixed sample and sequential search strategies

The behavior of a female in search of a mate determines the likelihood that she encounters a high-quality male in the search process. The fixed sample (best-of-n) search strategy and the sequential search (fixed threshold) strategy are two prominent models of search behavior. The sequential search strategy dominates the former strategy--yields an equal or higher expected net fitness return to searchers--when search costs are nontrivial and the distribution of quality among prospective mates is uniform or truncated normal. In this paper our objective is to determine whether there are any search costs or distributions of male quality for which the sequential search strategy is inferior to the fixed sample search strategy. The two search strategies are derived under general conditions in which females evaluate encountered males by inspection of an indicator character that has some functional relationship to male quality. The solutions are identical to the original models when the inspected male attribute is itself male quality. The sequential search strategy is shown to dominate the fixed sample search strategy for all search costs and distributions of male quality. Low search costs have been implicated to explain empirical observations that are consistent with the use of a fixed sample search strategy, but under conditions in which the original models were derived there is no search cost or distribution of male quality that favors the fixed sample search strategy. Plausible alternative explanations for the apparent use of this search strategy are discussed.

Chaotic unbounded differentiation operators

We construct dense sets of hypercyclic vectors for unbounded differention operators, including differentiation operators on the Hardy spaceH2, and the Laplacian operator onL2((Ω), for any bounded open subset Ω of ℝ2. Furthermore, we show that these operators are chaotic, in the sense of Devaney.

The variability of male quality and female mate choice decisions: second-order stochastic dominance and the behavior of searchers under a sequential search strategy.

The strategy employed by a female to sample prospective mates determines the likelihood that a high-quality male is encountered in the search process. In general, the choosiness of females is expected to depend on the variability of quality amongst the males that are sampled. The sequential search strategy is a prominent model of search behavior that involves the use of a threshold criterion to evaluate encountered individuals. In this paper, we show that the stochastic dominance of one distribution of male quality over another at the second order is necessary and sufficient for the optimal threshold criterion to differ under two distributions of male quality when the cost to sample males is held constant and the mean quality of males under each of the distributions is identical. A difference of the variance of male quality between two distributions does not imply that one distribution stochastically dominates the other at the second order and, hence, should not, in general, be used to assess the relative variability of quality amongst prospective mates. The adjustment of the threshold criterion in response to experimental manipulations of the distribution of male quality has been inferred from induced differences of the duration of search or the number of males sampled in the search process. Here we show that such inferences are unjustified. In particular, the difference of the threshold criterion imposed by second-order stochastic dominance does not determine the distribution under which females are expected to sample a larger number of males in the search process.

Cyclicity results for Jordan and compressed Toeplitz operators

A vectorx in a Hilbert spaceH iscyclic for a bounded linear operatorT∶H→H if the closed linear span of the orbit {Tnx∶n≥0} ofx underT is all ofH. Operators which have a cyclic vector are said to be cyclic.Jordan operators are the infinite direct sums of Jordan cells acting on finite- dimensional Hilbert spaces. Necessary and sufficient conditions for a Jordan operator to be cyclic are given (see Corollary 6). In this case, a dense set of cyclic vectors is exhibited (see Corollary 4). Sufficient conditions for uncountable collections of cyclic Jordan operators to have a common cyclic vector are given and, in this case, a dense set of common cyclic vectors is exhibited (see Corollary 9).Analogues of these cyclicity results for Jordan operators are obtained for compressions of analytic Toeplitz operatorsTA∶F→AF on the Hardy spaceH2 to subspaces (BH2)⊥ invariant for the backward shiftTz* whereB is a Blaschke product by showing that such compressions are quasisimilar to Jordan operators.

Mate choice and the evolutionary stability of a fixed threshold in a sequential search strategy

The sequential search strategy is a prominent model of searcher behavior, derived as a rule by which females might sample and choose a mate from a distribution of prospective partners. The strategy involves a threshold criterion against which prospective mates are evaluated. The optimal threshold depends on the attributes of prospective mates, which are likely to vary across generations or within the lifetime of searchers due to stochastic environmental events. The extent of this variability and the cost to acquire information on the distribution of the quality of prospective mates determine whether a learned or environmentally canalized threshold is likely to be favored. In this paper, we determine conditions on cross-generational perturbations of the distribution of male phenotypes that allow for the evolutionary stability of an environmentally canalized threshold. In particular, we derive conditions under which there is a genetically determined threshold that is optimal over an evolutionary time scale in comparison to any other unlearned threshold. These considerations also reveal a simple algorithm by which the threshold could be learned.

Reducing subspaces of compressed analytic Toeplitz operators on the Hardy space

In this paper we discuss necessary conditions and sufficient conditions for the compression of an analytic Toeplitz operator onto a shift coinvariant subspace to have nontrivial reducing subspaces. We give necessary and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial and obtain examples of reducing subspaces from these kernels. Motivated by this result we give necessary conditions and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial in terms of the supports of the two inner functions. By studying the commutant of a compression, we are able to give a necessary condition for the existence of reducing subspaces on certain shift coinvariant subspaces.

Semigroups of finite convolution operators

Abstract Semigroups of operators in the commutant A of the Volterra operator J: ƒ(x) → ∝ 0 x f(t) dt on L2(0, 1) are constructed and studied by means of a complex Fourier transform technique. A sufficient condition for a fixed operator in the algebra A to be embeddable in a C0 semigroup of operators in A is given. The condition is shown to be necessary under the further restriction that the semigroup consists of contraction operators (see Theorem 3). In this case, a description of all C0 semigroups of operators in A containing the fixed operator is obtained (see Theorem 2). Under mild conditions, the domain of analyticity of a semigroup of operators in A is shown to be a sector. Relationships between the lattices of closed invariant subspaces of the operators comprising a semigroup of operators in A and the resolvents of the infinitesimal generator of the semigroup are studied.

Semigroups of operators on quotient spaces of H2

Abstract Let φ be an inner function on the unit disc. Semigroups of operators in the commutant A (φ) of the compression of the standard unilateral shift onto H 2 φH 2 are constructed and studied using complex function theory. A sufficient condition for a fixed operator in the algebra A (φ) to be embeddable in a C0 semigroup of operators in A (φ) is given. The condition is shown to be necessary under the further restriction that the semigroup consists of contraction operators. Inner functions φ for which there exist operators in A (φ) which do not embed in a unique C0 semigroup of operators in A (φ) are constructed. Necessary and sufficient conditions for a coset in H ∞ φH ∞ to have a coset representative which is nonvanishing and bounded by one in modulus are given.