1,969 research outputs found
Quadratic forms and systems of forms in many variables
Let be quadratic forms with integer coefficients in
variables. When and the variety is a smooth
complete intersection, we prove an asymptotic formula for the number of integer
points in an expanding box at which these forms simultaneously vanish, which in
particular implies the Hasse principle for . Previous work
in this direction required to grow at least quadratically with . We give
a similar result for forms of degree , conditional on an upper bound for
the number of solutions to an auxiliary inequality. In principle this result
may apply as soon as . In the case that , several strategies
are available to prove the necessary upper bound for the auxiliary inequality.
In a forthcoming paper we use these ideas to apply the circle method to
nonsingular systems of forms with real coefficients.Comment: 29 pages, in revie
Systems of cubic forms in many variables
We consider a system of cubic forms in variables, with integer
coefficients, which define a smooth complete intersection in projective space.
Provided , we prove an asymptotic formula for the number of integer
points in an expanding box at which these forms simultaneously vanish. In
particular we can handle systems of forms in variables, previous work
having required that . One conjectures that should be
sufficient. We reduce the problem to an upper bound for the number of solutions
to a certain auxiliary inequality. To prove this bound we adapt a method of
Davenport.Comment: 23 pages, submitte
Atom Formation Rates Behind Shock Waves in Hydrogen and the Effect of Added Oxygen, July 1965 - July 1966
Formation rate of atomic hydrogen behind shock waves in hydrogen-argon mixture
Auctions with Heterogeneous Items and Budget Limits
We study individual rational, Pareto optimal, and incentive compatible
mechanisms for auctions with heterogeneous items and budget limits. For
multi-dimensional valuations we show that there can be no deterministic
mechanism with these properties for divisible items. We use this to show that
there can also be no randomized mechanism that achieves this for either
divisible or indivisible items. For single-dimensional valuations we show that
there can be no deterministic mechanism with these properties for indivisible
items, but that there is a randomized mechanism that achieves this for either
divisible or indivisible items. The impossibility results hold for public
budgets, while the mechanism allows private budgets, which is in both cases the
harder variant to show. While all positive results are polynomial-time
algorithms, all negative results hold independent of complexity considerations
On the Scaling Interpretation of Exponents in Hyperboloid Models of Delay and Probability Discounting
Previously, we (McKerchar et al., 2009) showed that two-parameter hyperboloid models (Green and Myerson, 2004; Rachlin, 2006) provide significantly better fits to delay discounting data than simple, one-parameter hyperbolic and exponential models. Here, we extend this effort by comparing fits of the two-parameter hyperboloid models to data from a larger sample of participants (N= 171) who discounted probabilistic as well as delayed rewards. In particular, we examined the effects of amount on the exponents in the two hyperboloid models of delay and probability discounting in order to evaluate key theoretical predictions of the standard psychophysical scaling interpretation of these exponents. Both the Rachlin model and the Green and Myerson model provided very good fits to delay and probability discounting of both small and large amounts at both the group and individual levels (all R2s \u3e .97 at the group level; all median R2s \u3e .92 at the individual level). For delay discounting, the exponent in both models did not vary as a function of delayed amount, consistent with the psychophysical scaling interpretation. For probability discounting, however, the exponent in both models increased as the probabilistic amount increased—a finding inconsistent with the scaling interpretatio
Budget Feasible Mechanisms for Experimental Design
In the classical experimental design setting, an experimenter E has access to
a population of potential experiment subjects , each
associated with a vector of features . Conducting an experiment
with subject reveals an unknown value to E. E typically assumes
some hypothetical relationship between 's and 's, e.g., , and estimates from experiments, e.g., through linear
regression. As a proxy for various practical constraints, E may select only a
subset of subjects on which to conduct the experiment.
We initiate the study of budgeted mechanisms for experimental design. In this
setting, E has a budget . Each subject declares an associated cost to be part of the experiment, and must be paid at least her cost. In
particular, the Experimental Design Problem (EDP) is to find a set of
subjects for the experiment that maximizes V(S) = \log\det(I_d+\sum_{i\in
S}x_i\T{x_i}) under the constraint ; our objective
function corresponds to the information gain in parameter that is
learned through linear regression methods, and is related to the so-called
-optimality criterion. Further, the subjects are strategic and may lie about
their costs.
We present a deterministic, polynomial time, budget feasible mechanism
scheme, that is approximately truthful and yields a constant factor
approximation to EDP. In particular, for any small and , we can construct a (12.98, )-approximate mechanism that is
-truthful and runs in polynomial time in both and
. We also establish that no truthful,
budget-feasible algorithms is possible within a factor 2 approximation, and
show how to generalize our approach to a wide class of learning problems,
beyond linear regression
Decoherence through Ancilla Anyon Reservoirs
We explore the decoherence of the gapless/critical boundary of a topological
order, through interactions with the bulk reservoir of "ancilla anyons." We
take the critical boundary of the toric code as an example. The intrinsic
nonlocal nature of the anyons demands the strong and weak symmetry condition
for the ordinary decoherence problem be extended to the strong or weak gauge
invariance conditions. We demonstrate that in the Hilbert
space, the partition function of the boundary is mapped to two layers of the
critical Ising model with an inter-layer line defect that depends on the
species of the anyons causing the decoherence. The line defects associated with
the tunneling of bosonic and anyons are relevant, and result in
long-range correlations for either the or anyon respectively on the
boundary in the doubled Hilbert space. In contrast, the defect of the anyon
is marginal and leads to a line of fixed points with varying effective central
charges, and power-law correlations having continuously varying scaling
dimensions. We also demonstrate that decoherence-analogues of Majorana zero
modes are localized at the spatial interface of the relevant and anyon
decoherence channels, which leads to a universal logarithmic scaling of the
R\'enyi entropy of the boundary.Comment: 6 + 5.5 page
Optimal Design of Robust Combinatorial Mechanisms for Substitutable Goods
In this paper we consider multidimensional mechanism design problem for
selling discrete substitutable items to a group of buyers. Previous work on
this problem mostly focus on stochastic description of valuations used by the
seller. However, in certain applications, no prior information regarding
buyers' preferences is known. To address this issue, we consider uncertain
valuations and formulate the problem in a robust optimization framework: the
objective is to minimize the maximum regret. For a special case of
revenue-maximizing pricing problem we present a solution method based on
mixed-integer linear programming formulation
Fixed Price Approximability of the Optimal Gain From Trade
Bilateral trade is a fundamental economic scenario comprising a strategically
acting buyer and seller, each holding valuations for the item, drawn from
publicly known distributions. A mechanism is supposed to facilitate trade
between these agents, if such trade is beneficial. It was recently shown that
the only mechanisms that are simultaneously DSIC, SBB, and ex-post IR, are
fixed price mechanisms, i.e., mechanisms that are parametrised by a price p,
and trade occurs if and only if the valuation of the buyer is at least p and
the valuation of the seller is at most p. The gain from trade is the increase
in welfare that results from applying a mechanism; here we study the gain from
trade achievable by fixed price mechanisms. We explore this question for both
the bilateral trade setting, and a double auction setting where there are
multiple buyers and sellers. We first identify a fixed price mechanism that
achieves a gain from trade of at least 2/r times the optimum, where r is the
probability that the seller's valuation does not exceed the buyer's valuation.
This extends a previous result by McAfee. Subsequently, we improve this
approximation factor in an asymptotic sense, by showing that a more
sophisticated rule for setting the fixed price results in an expected gain from
trade within a factor O(log(1/r)) of the optimal gain from trade. This is
asymptotically the best approximation factor possible. Lastly, we extend our
study of fixed price mechanisms to the double auction setting defined by a set
of multiple i.i.d. unit demand buyers, and i.i.d. unit supply sellers. We
present a fixed price mechanism that achieves a gain from trade that achieves
for all epsilon > 0 a gain from trade of at least (1-epsilon) times the
expected optimal gain from trade with probability 1 - 2/e^{#T epsilon^2 /2},
where #T is the expected number of trades resulting from the double auction
Eyewitness identification performance on showups improves with an additional-opportunities instruction: Evidence for present–absent criteria discrepancy
We tested the proposition that when eyewitnesses find it difficult to recognize a suspect (as in a culprit-absent showup), eyewitnesses accept a weaker match to memory for making an identification. We tie this proposition to the basic recognition memory literature, which shows people use lower decision criteria when recognition is made difficult so as to not miss their chance of getting a hit on the target. We randomly assigned participant–witnesses (N = 610) to a condition in which they were told that if they did not believe the suspect was the culprit, they would have additional opportunities to make an identification later (additional-opportunities instruction). We fully crossed this instruction with the standard admonition (i.e., the culprit may or may not be present) and with the presence or absence of the culprit in a showup identification procedure. The standard admonition had no impact on eyewitness decision-making; however, the additional-opportunities instruction reduced innocent-suspect identifications (from 33% to 15%) to a greater extent than culprit identifications (57% to 51%). The additional-opportunities instruction yielded a better tradeoff between culprit and innocent-suspect identifications as indicated by binary logistic regression and receiver operator characteristic (ROC) analyses
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