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As in earlier works, we consider {0,1}n as a sample space with a probability measure on it, thus making pseudo-Boolean functions into random variables. Under the assumption that the coordinate random variables are independent, we show it is very easy to give an orthonormal basis for the space of pseudo-Boolean random variables of degree at most k. We use this orthonormal basis to find the transform of a given pseudo-Boolean random variable and to answer various least squares minimization questions.

We consider {0,1}n as a sample space with a probability measure on it, thus making pseudo-Boolean functions into random variables. We then derive explicit formulas for approximating a pseudo-Boolean random variable by a linear function if the measure is permutation-invariant, and by a function of degree at most k if the measure is a product measure. These formulas generalize results due to Hammer–Holzman and Grabisch–Marichal–Roubens. We also derive a formula for the best faithful linear approximation that extends a result due to Charnes–Golany–Keane–Rousseau concerning generalized Shapley values. We show that a theorem of Hammer–Holzman that states that a pseudo-Boolean function and its best approximation of degree at most k have the same derivatives up to order k does not generalize to this setting for arbitrary probability measures, but does generalize if the probability measure is a product measure.

Computer security policies specify conditions for permissions to access various computer resources and information. Merging two security policies is needed when two organizations, together with their computer systems, merge into one entity as in corporate business acquisition. We propose a graph-theoretic method for merging the role/object hierarchies of two security policies. The formulation of merged hierachies is based on the graph minor relation in graph theory. Ideally, the merged role hierarchy should contain both the participating role hierarchies as graph minors, and similarly for the object hierarchy. We show that one can decide in polynomial time whether this ideal case is possible when the participating hierarchies are trees. We also show that in case the merged hierarchy exists, it can be constructed in polynomial time. Algorithms for detecting the feasibility of an ideal merged tree and for constructing the merged tree are presented. Our hierarchy/tree merge method is also applicable to the integration of heterogeneous databases with generalization hierarchies.

We study the problem of approximating pseudo-Boolean functions by linear pseudo-Boolean functions. Pseudo-Boolean functions generalize ordinary Boolean functions by allowing the function values to be real numbers instead of just the 0-1 values. Pseudo-Boolean functions have been used by AI and theorem proving researchers for efficient constraint satisfaction solving. They can also be applied for modeling uncertainty. We investigate the possibility of efficiently computing a linear approximation of a pseudo-Boolean function of arbitrary degree. We show some example cases in which a simple (efficiently computable) linear approximating function works just as well as the best linear approximating function, which may require an exponential amount of computation to obtain. We conjecture that for any pseudo-Boolean function of fixed degree k >1 where k is independent of the number of Boolean variables, the best linear approximating function works better than simply using the linear part of the target function. We also study the behavior of the expected best linear approximating function when the target pseudo-Boolean function to be approximated is random.

We consider anonymous secure communication, where parties not only wish to conceal their communications from outside observers, but also wish to conceal the very fact that they are communicating. We consider the bus framework introduced by Beimel and Dolev (J. Cryptology 16 (2003) 25), where messages are delivered by a bus traveling on a random walk. We generalize this idea to consider more than one bus. We show that if w buses are allowed, then the expected delivery time for a message can be decreased from Θ(n) to Θ(n/w) in the case of a complete graph. Additionally, we introduce a class of graphs called r-partite directed collars and obtain analogous bounds on the expected delivery time for these graphs. We also propose several new features that resolve possible shortcomings in the systems proposed by Beimel and Dolev.

We define a class of codes that we call affine variety codes. These codes are obtained by evaluating functions in the coordinate ring of an affine variety on all the F_q-rational points of the variety. We show that one can, at least in theory, decode these codes up to half the true minimum distance by using the theory of Grobner bases. We extend results of A. B. Cooper and of X. Chen, I. S. Reed, T. Helleseth, and T. K. Truong.

Let X X denote the rational curve with n + 1 n+1 nodes obtained from the Riemann sphere by identifying 0 with ∞ \infty and ζ j \zeta ^j with − ζ j -\zeta ^j for j = 0 , 1 , … , n − 1 j=0,1,\dots ,n-1 , where ζ \zeta is a primitive ( 2 n ) (2n) th root of unity. We show that if n n is even, then X X has no smooth Weierstrass points, while if n n is odd, then X X has 2 n 2n smooth Weierstrass points.

We prove that if there are consecutive gaps at a rational point on a smooth curve defined over a finite field, then one can improve the usual lower bound on the minimum distance of certain algebraic-geometric codes defined using a multiple of the point.

We study the problem of detecting and profiling terrorists using a combination of ordinary flat classifiers and relational information. Our starting point is a database for a set of individuals characterized by both ldquolocalrdquo attributes such as age and criminal background, and ldquorelationalrdquo information such as communications among a subset of the individuals. A subset of the individuals have labels (terrorist or normal people), and we would want to design a classifier that captures the patterns of terrorists and achieves good accuracy in predicting the labels of the remaining part of the database. While ldquoflatrdquo (or attribute-based) classifiers such as decision trees mainly concentrate on using local attributes for object classification, the ldquoguilty by associationrdquo simple relational classifier utilizes only connections (relations) among objects for the classification task. We present a hybrid approach that iteratively applies a flat classifier augmented with flattened relational attributes for learning and classification. In particular, we describe our experiments on combining fuzzy C-means clustering with iterative relational classification for terrorist detection from the database. Our hybrid classifier has achieved very good prediction accuracy in the experiments.

We introduce local soft belief updating, a new heuristic for taking into account relations that exist between entities in a database. Our idea applies Pearl's belief updating, but only in the first-order neighborhood of each node, thus avoiding any problems with loops. We apply our method to a classification problem using a subset of the Cora database of computer science articles, with Cora's citation graph giving the relations between entities.

We recently proposed the Edgewise Greedy Algorithm (EGA) for learning a decomposable Markov network of treewidth k approximating a given joint probability distribution of n discrete random variables. The main ingredient of our algorithm is the stepwise forward selection algorithm (FSA) due to Deshpande, Garofalakis, and Jordan. EGA is an efficient alternative to the algorithm (HGA) by Malvestuto, which constructs a model of treewidth k by selecting hyperedges of order k+1. In this paper, we present results of empirical studies that compare HGA, EGA and FSA-K which is a straightforward application of FSA, in terms of approximation accuracy (measured by KL-divergence) and computational time. Our experiments show that (1) on the average, all three algorithms produce similar approximation accuracy; (2) EGA produces comparable or better approximation accuracy and is the most efficient among the three. (3) Malvestuto's algorithm is the least efficient one, although it tends to produce better accuracy when the treewidth is bigger than half of the number of random variabls; (4) EGA coupled with local search has the best approximation accuracy overall, at a cost of increased computation time by 50 percent.

Let Y denote a smooth, projective curve of genus g defined over C. A point P G Y is a Weierstrass point if there exists a rational function on Y with a pole only at P of order at most g, or if there exists a regular differential on Y which vanishes at P to order at least g, or if the divisor gP is special. However, the most "functorial" way to define Weierstrass points is as the zeros of the wronskian, a section of the (g(g + l)/2)th tensor power of the canonical bundle on Y. It is this last definition that we use as the foundation for defining Weierstrass points on singular curves. What is essential is that the sheaf of dualizing differentials should be locally free and this is exactly the property satisfied by Gorenstein curves. Let X be an integral, projective Gorenstein curve of arithmetic genus g > 0 over C. Let ω denote the bundle of dualizing differentials on X and let Sf denote an invertible sheaf on X. Put s = dim H°(X,J?) = h°(&). Assume s > 0 and choose a basis φΪ9...9φsfoτH0(X,5?). We will define a section of S?®sΘω^s'1^2 as follows: Suppose that {U^} is a covering of X by open subsets such that ^f(U^) and ω(UM) are free ^(^ (α) )-modules generated by ψ(a) and τ^a\ respectively. Define FJ f e YilJ^^x) inductively

In this paper, we present a new framework for learning pseudo-Boolean functions from training data. The new learning framework is based on the observation that the training data can be seen as constraints on the possible candidate pseudo-Boolean functions and that, without any additional information, any of the pseudo-Boolean functions satisfying these constraints is equally likely. We define two types of learning from a training data set: one is to learn to predict the probability that the target function value f*(x/sub 0/) falls within an interval [a, b]; and the other is to learn a specific pseudo-Boolean function f as an approximation of the target function f*. Efficient algorithms for both learning tasks are presented. We relate our approach to the Bayesian classifier method. We argue that the new learning framework is suitable for applications in which the training data is rather limited and yet one would like to make useful and reliable predictions on the future data points. Applications in terrorist detection and classification clearly present such a situation where training data for terrorists are rather scarce.

Pseudo-Boolean functions are generalizations of Boolean functions. We present a new method for learning pseudo-Boolean functions from limited training data. The objective of learning is to obtain a function f which is a good approximation of the target function f*. We define suitable criteria for the “goodness” of an approximating function. One criterion is to choose a function f that minimizes the “expected distance” with respect to a distance function d (over pairs of pseudo-Boolean functions) and the uniform distribution over all feasible pseudo-Boolean functions. We define two alternative “distance measures” over pairs of pseudo-Boolean functions, and show that they are are actually equivalent with respect to the criterion of minimal expected distance. We outline efficient algorithms for learning pseudo-Boolean functions according to these criteria. Other reasonable distance measures and “goodness” criteria are also discussed.

This is a clinical paper, which includes material from sessions, presenting the process of the analysis of a young adult male whose narcissistic character patterns, related to and evolved from failed attempts to integrate conflicting parental identifications. These unintegrated mother and father identifications contributed to life-long latent homosexual fantasies. Rodney's analysis indicates that for a boy/man, even a mother who has the qualities of a good enough parent, is not good enough to enable him to reach a nonconflicted manhood. A mother cannot provide for the boy the male model he needs and is searching for, the male who would affirm him in his maleness. Rodney wanted a father on whose shoulders he could stand to become a man. Rodney's persistent homosexual fantasy life, his quest for father's love, and his search for a masculine object to identify with stem from a combination of several factors. Rodney's regression to the negative oedipal phase was probably stimulated when father left the family. Rodney was eleven at the time. He felt overwhelmed experiencing himself as the oedipal victor. Unconsciously, Rodney feared his exacerbated incestuous wishes. He projected them upon his mother and subsequently incorporated them in his fantasies. His regression to more infantile dependency feelings was defensive. Rodney's father was an unsuitable object for identification. He was disinterested in Rodney and emotionally unavailable. Rodney, however, sought his father, whose lack of loving acknowledgement resulted in a lack of affirmation of Rodney's masculinity. Mother provided for Rodney the loving acknowledgement he lacked in his relationship with father. She was emotionally sustaining, an energetic, vibrant personality, who was seen by Rodney as a "superior human being." Rodney consciously idealized his mother toward whom he unconsciously also had ambivalent feelings. Rodney's identification with mother was not counterbalanced by the presence of a strong, loving father figure whom he could have used as a suitable model. This led to the development in Rodney of a strong sense of effeminization. Rodney in his homosexual fantasies assumed the so-called "feminine victimized" role. The regression to the negative oedipal phase contributed to an exacerbation of erotic, father-directed feelings, intensified by the identification with mother. Rodney was fixated in his quest for father's love. In addition, Rodney's unconscious guilt related to father and mother directed incestuous impulses, and his intense aggressivesadistic feelings contributed to the masochistic cast of his masturbation fantasies. Rodney's narcissistic aims and the quality of his narcissism changed during the analysis. His grandiosity almost disappeared. Rodney's goals became realistic and he acquired the skills necessary to achieve them. Inhibitions related to the "fear of success" were worked through. This enabled Rodney to compete successfully. His healthy narcissism derives from the success of his many achievements. Though Rodney remained a basically narcissistic personality, he did derive great pleasure from being a giving person. This was one of the many ways in which he identified with his mother. At the present, Rodney's identifications are selective and do not evoke intrapsychic conflict.

The author examines the development of the girl's psychosexual feminine self-image during the preoedipal and early oedipal phases, as well as the accompanying primary and secondary genital feelings and anxieties. The connection between the little girl's discovery and pleasure in self-exploration and masturbation and related anxieties about being "closed-up" is discussed. The nature of the corresponding object relationships is elucidated, especially the narcissistic significance of being like mother.

1. Let X denote a nonsingular, geometrically irreducible, projective curve of genus g defined over the finite field Fq with q elements. Assume that X has Fq-rational points. Let D be a divisor on X defined over Fq (i.e., D is invariant under Gal(F-~q/Fq)). Then L ( D ) will denote the Fq-vector space of all rational functions f on X, defined over Fq, with divisor (f) > D , together with the zero function, and f~(D) will denote the Fqvector space of all rational differentials 77 on X, defined over Fq, with divisor (~/) > D, together with the zero differential. Put l (D) = dimfqL(D) and i (D) = dimFqf~(D). The Riemann-Roch Theorem states that

Motivated by narcissistic and libidinal strivings in a pathological family setting, the patient developed discordant identification systems which neither fused nor modulated each other. The structure of her wished-for self image with its autonomous tendencies was based on an Older Brother fantasy with which she identified. Her submissive self image was determined by an unconscious ideal modeled on her mother. These incompatible identifications continued in conflict. Enactment of the pattern determined by the maternal ideal led to object choices which evoked masochistic behavior, recognized by the patient as destructive to her ego aims.

Let T g {T_g} denote the Teichmüller space and let V V denote the universal family of Teichmüller surfaces of genus g g Let V T g ( n ) V_{{T_g}}^{(n)} denote the n n th symmetric product of V V over T g {T_g} and let J J denote the family of Jacobians over T g {T_g} . Let f : V T g ( n ) → J f:V_{{T_g}}^{(n)} \to \text {J} be the natural relativization over T g {T_g} of the classical map defined by integrating holomorphic differentials. Let \[ u : f ∗ Ω J / T g 1 → Ω V T g / T g ( n ) 1 u:{f^\ast }\Omega _{\text {J} /{T_g}}^1 \to \Omega _{V_{{T_g}/{T_g}}^{(n)}}^1 \] be the map induced by f f . We define G n r G_n^r to be the analytic subspace of V T g ( n ) V_{{T_g}}^{(n)} defined by the vanishing of ∧ n − r + 1 u { \wedge ^{n - r + 1}}u . Put τ = ( r + 1 ) ( n − r ) − r g \tau = (r + 1)(n - r) - rg . We show that G n 1 − G n 2 G_n^1 - G_n^2 , if nonempty, is smooth of pure dimension 3 g − 3 + τ + 1 3g - 3 + \tau + 1 . From this result, we may conclude that, for a generic curve X X , the fiber of G n 1 − G n 2 G_n^1 - G_n^2 over the module point of X X , if nonempty, is smooth of pure dimension τ + 1 \tau + 1 , a classical assertion. Variational formulas due to Schiffer and Spencer and Rauch are employed in the study of G n r G_n^r .

Complex spacss W n r \mathcal {W}_n^r of Weierstrass points are isomorphic to the intersection, on the n n th symmetric product of the universal curve over the Teichmüller space, of complex spaces G n r \mathcal {G}_n^r of special divisors with the diagonal Δ n {\Delta _n} consisting of divisors which are multiples of a point. The tangent space at a point of this intersection is described and it is shown that G n 1 − G n 2 \mathcal {G}_n^1 - \mathcal {G}_n^2 and Δ n {\Delta _n} intersect transversally.

We determine, except for one unsettled case, which combinations of Weierstrass weights can occur on irreducible rational nodal curves of arithmetic genus three. It is shown that the number of nonsingular Weierstrass points on such curves can be any integer between 0 and 6, except 1.

C. Widland [14] has defined Weierstrass points on integral, projective Gorenstein curves. We show here that the Weierstrass points on a generic integral rational nodal curve have the minimal possible weights or, equivalently, that such a curve has the maximum possible number of distinct nonsingular Weierstrass points. Rational curves with g nodes arise in degeneration arguments involving smooth curves of genus g and they have also recently arisen in connection with g-soliton solutions to certain nonlinear partial differential equations [11], [13].

Weierstrass points are defined for invertible sheaves on integral, projective Gorenstein curves. An example is given of a rational nodal curveX and an invertible sheaf ℒ of positive degree onX such that the set of all higher order Weierstrass points of ℒ is not dense inX.

Let X denote a compact Riemann surface of genus g > 1. At each point P~X, there is a sequence of g integers, l=71(P) g. Every compact Riemann surface of genus > 1 has a finite number of Weierstrass points. (See [6] for details.) The question of describing the moduli of compact Riemann surfaces of genus g which have a Weierstrass point with a specified first nongap has been answered by Rauch [14] and Arbarello [1]. The problem of describing the moduli of curves of genus g which have a Weierstrass point with a specified gap sequence has been considered by Hensel-Landsberg [7] and Rauch [15] and recently results on this question have been obtained by Pinkham [13] and RimVitulli [16]. In [9], we defined complex spaces of Weierstrass points of the universal curve and obtained a theorem concerning the smoothness and dimension of these spaces. From this, one can recover the moduli results of Rauch [14]. In order to study the higher Weierstrass nongaps and the entire gap sequence, we must extend our definition in [9]. We do this in w and then prove a theorem concerning the smoothness and dimension of these spaces which is analogous to the main result in [9]. F rom this, we can recover a moduli result of Pinkham [13]. In w 3, we study the geometry of these complex spaces of Weierstrass points of the universal curve of genus 4. We obtain a description of the moduli of Teichmiiller surfaces of genus 4 which have a Weierstrass point with a specified gap sequence. In the following, a complex space is not necessarily reduced. If Y is a complex space, then IYI will denote the underlying set of points. If Y, Z1 . . . . . Z , are closed complex subspaces of a complex space, then by Y Z 1 Z 2 . . . Z

The classical theory of Weierstrass points on a compact Riemann surface is well-known (see, for example, [3]). Ogawa [6] has defined generalized Weierstrass points. Let Y denote a compact complex manifold of (complex) dimension n. Let E denote a holomorphic vector bundle on Y of rank q. Let Jk(E) (k = 0, 1, …) denote the holomorphic vector bundle of k-jets of E [2, p. 112]. Put rk(E) = rank Jk(E) = q.(n + k)!/n!k!. Suppose that Γ(E), the vector space of global holomorphic sections of E, is of dimension γ(E)>0. Consider the trivial bundle Y × Γ(E) and the mapwhich at a point Q∈Y takes a section of E to its k-jet at Q. Put μ = min(γ(E),rk(E)).

We extend results of K. Lee and M.E. OʼSullivan by showing how to use Grobner bases to find the interpolation polynomial for list decoding a one-point AG code C = C L ( r P , D ) on any curve X , where P is an F q -rational point on X and D = P 1 + P 2 + ⋯ + P n is the sum of other F q -rational points on X . We then define the generic interpolation polynomial for list decoding such a code. The generic interpolation polynomial should specialize to the interpolation polynomial for most received strings. We give an example of a family of Reed–Solomon 1-error correcting codes for which a single error can be decoded by a very simple process involving substituting into the generic interpolation polynomial.

Let F"q denote the finite field with q elements. The symmetric group S"n acts on the set G"n","q of all linear subspaces of F"q^n by permuting coordinates. We show that the normalized character of the associated permutation representation of S"n approaches the trivial character as n approaches infinity. We also consider the case of the wreath product of F"q^x and S"n acting on G"n","q, and formulate a conjecture that would yield the asymptotic number of distinct equivalence classes of q-ary linear codes.

It is shown that as a family of projective smooth curves degenerates to an irreducible Gorenstein curve the Weierstrass weight at a point P on the limit curve is the sum of the Weierstrass weights at points on the smooth curves converging to P. C. Widland [8] extended the classical notion of Weierstrass point to integral, projective Gorenstein curves. We considered Weierstrass points of invertible sheaves on such curves in [5] and showed that a singular point is always a Weierstrass point of high weight of any invertible sheaf with at least two linearly independent global sections. We remarked that this may be interpreted to mean that as a family of smooth curves degenerates to an irreducible Gorenstein curve, then many of the Weierstrass points tend towards the singularities. Our goal in this note is to make this statement precise. We work over C. By a "point" of a scheme, we mean a closed point. If X is a scheme of finite type over C, then we denote the associated complex space by Xan. We wish to thank Augusto Nobile and Peter Stiller for helpful conversations. Let S be an integral, noetherian scheme over C and let ir: X -S be a family of Gorenstein curves of arithmetic genus g > 2. By this we will mean that 7r is a flat, proper morphism whose geometric fibers are integral Gorenstein curves of arithmetic genus g. By the theory of duality of coherent sheaves [4, V.9.7 and VII.4 (p. 388)], there is a canonical invertible sheaf w = wx/s on X whose restriction to a fiber X, is the sheaf of dualizing differentials on that curve. Let L be an invertible sheaf on X. Then L is flat over S. Assume that dim h0(X,, Ls) = r > 0 for all s E S. Then 7r2 is locally free of rank r. We define below an effective Cartier divisor 1 (L) on X. We will also use 1 (L) to denote the associated locally principal closed subscheme of X. We note that the subscheme W(L) may not be reduced. The points of W(L) are the L-Weierstrass points of X/S. The wOn-Weierstrass points are called Weierstrass points of order n and the w-Weierstrass points are simply called Weierstrass points. Suppose (s, P) E X and let U be an open neighborhood of (s, P) such that (1) There are sectionsl, ,. . ., Or of L whose restrictions to X,, form a basis for HO(X5,, , L) for all s' e ir(U). (2) LC(U) is a free Ox (U)-module generated by /. (3) wx/s (U) is a free Ox (U)-module generated by r. Define F%,j E F(U, OX) by q$3 = F1,> for j=,...,r, dFi-1,3 = Ft,,3 for i = 2, . ,r and j = 1,. . . ., r, Received by the editors May 15, 1986 and, in revised form, June 27, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 14H15.

An analysis of the structural and defensive aspects of the self-righteous personality has been presented. Some genetic factors leading to the development of such a personality were considered. Analysis indicated that introjection of the idealized, righteous parent and identification with this introject was motivated by the need for narcissistic supplies. Fusion with this introject occurs during outbursts of self-righteous indignation directed towards an object cathected to a great extent with narcissistic libido. For the unconscious of the patient such an object represents in large measure the repressed aspects of his devalued self-representation. Some dynamic constellations encountered during treatment have been described.

The developmental history of pathological traits and constellations, specifically their origin in the context of the parent-child relationship, their initially adaptive nature derived from parental valuations as well as the role of identification with the aggressor, has been described. The narcissistic investment of these traits is accounted for by these factors. With character neurotics, in the first phase of treatment, the analysis of contradictions brings about a state of psychic disequilibrium. The patient, in this process, experiences a loss of self-esteem. This is related to the discovery of the nature of his unconcious motivations, to the painful awareness of his limitations, and the recognition that participation in parental omnipotence is an illusion. These insights result in a temporary narcissistic depression. Intrapsychic separation from the pathogenic introject, combined with the internalization of the analytic attitudes of empathy and tolerance, leads to the modulation of the pathological ego ideal. Mourning, which accompanies the renunciation of one's fantasied grandiose self, eventuates in the development, by the patient, of a capacity of empathy with himself and the formation of a more mature and realistic wishful self-image.