multiple kernel learning for dimensionality reduction in solving complex visual learning tasks adopting multiple descriptors to more precisely characterize the data has been a feasible way for improving performance the resulting data representations are typically high dimensional and assume diverse forms hence finding a way of transforming them into a unified space of lower dimension generally facilitates the underlying tasks such as object recognition or clustering to this end the proposed approach termed mkl dr generalizes the framework of multiple kernel learning for dimensionality reduction and distinguishes itself with the following three main contributions first our method provides the convenience of using diverse image descriptors to describe useful characteristics of various aspects about the underlying data second it extends a broad set of existing dimensionality reduction techniques to consider multiple kernel learning and consequently improves their effectiveness third by focusing on the techniques pertaining to dimensionality reduction the formulation introduces a new class of applications with the multiple kernel learning framework to address not only the supervised learning problems but also the unsupervised and semi supervised ones

exact partition functions for potts antiferromagnets on cyclic lattice strips abstract we present exact calculations of the zero temperature partition function of the q state potts antiferromagnet on arbitrarily long strips of the square triangular and kagome lattices with width l y 2 or 3 vertices and with periodic longitudinal boundary conditions from these in the limit of infinite length we obtain the exact ground state entropy s 0 k b ln w these results are of interest since this model exhibits nonzero ground state entropy s 0 0 for sufficiently large q and hence is an exception to the third law of thermodynamics we also include results for homeomorphic expansions of the square lattice strip the analytic properties of w q are determined and related to zeros of the chromatic polynomial for long finite strips

ground state entropy of the potts antiferromagnet on strips of the square lattice abstract we present exact solutions for the zero temperature partition function chromatic polynomial p and the ground state degeneracy per site w exponent of the ground state entropy for the q state potts antiferromagnet on strips of the square lattice of width ly vertices and arbitrarily great length lx vertices the specific solutions are for a ly 4 fbcy pbcx cyclic b ly 4 fbcy tpbcx mobius c ly 5 6 pbcy fbcx cylindrical and d l y 5 fbc y fbc x open where fbc pbc and tpbc denote free periodic and twisted periodic boundary conditions respectively in the lx limit of each strip we discuss the analytic structure of w in the complex q plane the respective w functions are evaluated numerically for various values of q several inferences are presented for the chromatic polynomials and analytic structure of w for lattice strips with arbitrarily great ly the absence of a nonpathological lx limit for real nonintegral q in the interval 0 0 for strips of the square triangular lattice is discussed

contextual object localization with multiple kernel nearest neighbor recently many object localization models have shown that incorporating contextual cues can greatly improve accuracy over using appearance features alone therefore many of these models have explored different types of contextual sources but only considering one level of contextual interaction at the time thus what context could truly contribute to object localization through integrating cues from all levels simultaneously remains an open question moreover the relative importance of the different contextual levels and appearance features across different object classes remains to be explored here we introduce a novel framework for multiple class object localization that incorporates different levels of contextual interactions we study contextual interactions at the pixel region and object level based upon three different sources of context semantic boundary support and contextual neighborhoods our framework learns a single similarity metric from multiple kernels combining pixel and region interactions with appearance features and then applies a conditional random field to incorporate object level interactions to effectively integrate different types of feature descriptions we extend the large margin nearest neighbor to a novel algorithm that supports multiple kernels we perform experiments on three challenging image databases graz 02 msrc and pascal voc 2007 experimental results show that our model outperforms current state of the art contextual frameworks and reveals individual contributions for each contextual interaction level as well as appearance features indicating their relative importance for object localization

chromatic polynomials for j h i strip graphs and their asymptotic limits abstract we calculate the chromatic polynomials p for n vertex strip graphs of the form j l 1 m h i where j and i are various subgraphs on the left and right ends of the strip whose bulk is comprised of m fold repetitions of a subgraph h the strips have free boundary conditions in the longitudinal direction and free or periodic boundary conditions in the transverse direction this extends our earlier calculations for strip graphs of the form l 1 m h i we use a generating function method from these results we compute the asymptotic limiting function w lim n p 1 n for q z this has physical significance as the ground state degeneracy per site exponent of the ground state entropy of the q state potts antiferromagnet on the given strip in the complex q plane w is an analytic function except on a certain continuous locus b in contrast to the l 1 m h i strip graphs where b i is independent of i and ii consists of arcs and possible line segments that do not enclose any regions in the q plane we find that for some j l 1 m h i strip graphs b i does depend on i and j and ii can enclose regions in the q plane our study elucidates the effects of different end subgraphs i and j and of boundary conditions on the infinite length limit of the strip graphs

asymptotic limits and zeros of chromatic polynomials and ground state entropy of potts antiferromagnets we study the asymptotic limiting function w g q lim n to infty p g q 1 n where p g q is the chromatic polynomial for a graph g with n vertices we first discuss a subtlety in the definition of w g q resulting from the fact that at certain special points q s the following limits do not commute lim n to infty lim q to q s p g q 1 n ne lim q to q s lim n to infty p g q 1 n we then present exact calculations of w g q and determine the corresponding analytic structure in the complex q plane for a number of families of graphs g including circuits wheels biwheels bipyramids and cyclic and twisted ladders we study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs all but a finite number of the zeros lie exactly on a unit circle whose position depends on the family using the connection of p g q with the zero temperature potts antiferromagnet we derive a theorem concerning the maximal finite real point of non analyticity in w g q denoted q c and apply this theorem to deduce that q c sq 3 and q c hc 3 sqrt 5 2 for the square and honeycomb lattices finally numerical calculations of w hc q and w sq q are presented and compared with series expansions and bounds

oscillatory critical amplitudes in hierarchical models we study the oscillatory critical amplitudes of theq states potts model on a diamond hierarchical lattice we consider an example of the generic case finite critical index as well as the degenerate case essential singularity in both cases we compare the magnitude of the oscillations with geometrical characteristics of the julia set of zeroes of the partition function

exact potts model partition functions on wider arbitrary length strips of the square lattice abstract we present exact calculations of the partition function of the q state potts model for general q and temperature on strips of the square lattice of width l y 3 vertices and arbitrary length l x with periodic longitudinal boundary conditions of the following types i fbc y pbc x cyclic ii fbc y tpbc x mobius iii pbc y pbc x toroidal and iv pbc y tpbc x klein bottle where fbc and t pbc refer to free and twisted periodic boundary conditions results for the l y 2 torus and klein bottle strips are also included in the infinite length limit the thermodynamic properties are discussed and some general results are given for low temperature behavior on strips of arbitrarily great width we determine the submanifold in the c 2 space of q and temperature where the free energy is singular for these strips our calculations are also used to compute certain quantities of graph theoretic interest

multik mhks a novel multiple kernel learning algorithm in this paper we develop a new effective multiple kernel learning algorithm first we map the input data into m different feature spaces by m empirical kernels where each generated feature space is taken as one view of the input space then through borrowing the motivating argument from canonical correlation analysis cca that can maximally correlate the m views in the transformed coordinates we introduce a special term called inter function similarity loss rifsi into the existing regularization framework so as to guarantee the agreement of multiview outputs in implementation we select the modification of ho kashyap algorithm with squared approximation of the misclassification errors mhks as the incorporated paradigm and the experimental results on benchmark data sets demonstrate the feasibility and effectiveness of the proposed algorithm named multik mhks

object detection with dog scale space a multiple kernel learning approach difference of gaussians dog scale space for an image is a significant way to generate features for object detection and classification while applying dog scale space features for object detection classification we face two inevitable issues dealing with high dimensional data and selecting weighting of proper scales the scale selection process is mostly ad hoc to present in this paper we propose a multiple kernel learning mkl method for both dog scale selection weighting and dealing with high dimensional scale space data we design a novel shift invariant kernel function for dog scale space to select only the useful scales in the dog scale space a novel framework of mkl is also proposed we utilize a 1 norm support vector machine svm in the mkl optimization problem for sparse weighting of scales from dog scale space the optimized data dependent kernel accommodates only a few scales that are most discriminatory according to the large margin principle with a 2 norm svm this learned kernel is applied to a challenging detection problem in oil sand mining to detect large lumps in oil sand videos we tested our method on several challenging oil sand data sets our method yields encouraging results on these difficult to process images and compares favorably against other popular multiple kernel methods

log polar wavelet energy signatures for rotation and scale invariant texture classification classification of texture images is important in image analysis and classification this paper proposes an effective scheme for rotation and scale invariant texture classification using log polar wavelet signatures the rotation and scale invariant feature extraction for a given image involves applying a log polar transform to eliminate the rotation and scale effects but at same time produce a row shifted log polar image which is then passed to an adaptive row shift invariant wavelet packet transform to eliminate the row shift effects so the output wavelet coefficients are rotation and scale invariant the adaptive row shift invariant wavelet packet transform is quite efficient with only o n spl middot log n complexity a feature vector of the most dominant log polar wavelet energy signatures extracted from each subband of wavelet coefficients is constructed for rotation and scale invariant texture classification in the experiments we employed a mahalanobis classifier to classify a set of 25 distinct natural textures selected from the brodatz album the experimental results based on different testing data sets for images with different orientations and scales show that the proposed classification scheme using log polar wavelet signatures outperforms two other texture classification methods its overall accuracy rate for joint rotation and scale invariance being 90 8 percent demonstrating that the extracted energy signatures are effective rotation and scale invariant features concerning its robustness to noise the classification scheme also performs better than the other methods

dictionary learning for stereo image representation one of the major challenges in multi view imaging is the definition of a representation that reveals the intrinsic geometry of the visual information sparse image representations with overcomplete geometric dictionaries offer a way to efficiently approximate these images such that the multi view geometric structure becomes explicit in the representation however the choice of a good dictionary in this case is far from obvious we propose a new method for learning overcomplete dictionaries that are adapted to the joint representation of stereo images we first formulate a sparse stereo image model where the multi view correlation is described by local geometric transforms of dictionary elements atoms in two stereo views a maximum likelihood ml method for learning stereo dictionaries is then proposed where a multi view geometry constraint is included in the probabilistic model the ml objective function is optimized using the expectation maximization algorithm we apply the learning algorithm to the case of omnidirectional images where we learn scales of atoms in a parametric dictionary the resulting dictionaries provide better performance in the joint representation of stereo omnidirectional images as well as improved multi view feature matching we finally discuss and demonstrate the benefits of dictionary learning for distributed scene representation and camera pose estimation

scale affine invariant interest point detectors in this paper we propose a novel approach for detecting interest points invariant to scale and affine transformations our scale and affine invariant detectors are based on the following recent results 1 interest points extracted with the harris detector can be adapted to affine transformations and give repeatable results geometrically stable 2 the characteristic scale of a local structure is indicated by a local extremum over scale of normalized derivatives the laplacian 3 the affine shape of a point neighborhood is estimated based on the second moment matrix our scale invariant detector computes a multi scale representation for the harris interest point detector and then selects points at which a local measure the laplacian is maximal over scales this provides a set of distinctive points which are invariant to scale rotation and translation as well as robust to illumination changes and limited changes of viewpoint the characteristic scale determines a scale invariant region for each point we extend the scale invariant detector to affine invariance by estimating the affine shape of a point neighborhood an iterative algorithm modifies location scale and neighborhood of each point and converges to affine invariant points this method can deal with significant affine transformations including large scale changes the characteristic scale and the affine shape of neighborhood determine an affine invariant region for each point we present a comparative evaluation of different detectors and show that our approach provides better results than existing methods the performance of our detector is also confirmed by excellent matching resultss the image is described by a set of scale affine invariant descriptors computed on the regions associated with our points

scale invariant features and polar descriptors in omnidirectional imaging we propose a method to compute scale invariant features in omnidirectional images we present a formulation based on the riemannian geometry for the definition of differential operators on non euclidian manifolds that adapt to the mirror and lens structures in omnidirectional imaging these operators lead to a scale space analysis that preserves the geometry of the visual information in omnidirectional images we then build a novel scale invariant feature detection framework for omnidirectional images that can be mapped on the sphere we further present a new descriptor and feature matching solution for these omnidirectional images the descriptor builds on the log polar planar descriptors and adapts the descriptor computation to the specific geometry and the nonuniform sampling density of omnidirectional images we also propose a rotation invariant matching method that eliminates the orientation computation during the feature detection phase and thus decreases the computational complexity experimental results demonstrate that the new feature computation method combined with the adapted descriptors offers promising detection and matching performance i e it improves on the common scale invariant feature transform sift features computed on the unwrapped omnidirectional images as well as spherical sift features finally we show that the proposed framework also permits to match features between images with different native geometry

phase transitions on fractals iii infinitely ramified lattices for pt ii see ibid vol 17 p 435 1984 in the first two papers of this series the authors considered self similar fractal lattices with a finite order of ramification r in the present paper they study physical models defined on a family of fractals with r infinity in order to characterise the geometry of these systems they need the connectivity q and the lacunarity l in addition to the fractal dimensionality d it is found that discrete symmetry spin models on these lattices undergo a phase transition at tc 0 an approximate renormalisation group scheme is constructed and used to find the dependence of tc and the critical exponents on the geometrical factors they also solve the problem of resistor networks on these fractals and discuss its consequences concerning spin models with continuous symmetry

asymptotics of the susceptibility for the ising model on the hierarchical lattices using the gks inequality griffiths and kaufman showed in 12 that the susceptibility in the ising model on the hierarchical lattices is infinite fort tc in the present work we give an analytic calculation of the dominant singularity of the susceptibility whenh 0 fort tc moreover we analytically find the spontaneous magnetization fort tc and prove that fort tc the susceptibility is finite

chromatic polynomials for families of strip graphs and their asymptotic limits abstract we calculate the chromatic polynomials p gs m q and from these the asymptotic limiting functions w g s q lim n p g s q 1 n for families of n vertex graphs gs m comprised of m repeated subgraphs h adjoined to an initial graph i these calculations of w gs q for infinitely long strips of varying widths yield important insights into properties of w q for two dimensional lattices in turn these results connect with statistical mechanics since w q is the ground state degeneracy of the q state potts model on the lattice for our calculations we develop and use a generating function method which enables us to determine both the chromatic polynomials of finite strip graphs and the resultant w gs q function in the limit n from this we obtain the exact continuous locus of points b where w gs q is nonanalytic in the complex q plane this locus is shown to consist of arcs which do not separate the q plane into disconnected regions zeros of chromatic polynomials are computed for finite strips and compared with the exact locus of singularities b we find that as the width of the infinitely long strips is increased the arcs comprising b elongate and move toward each other which enables one to understand the origin of closed regions that result for the infinite 2d lattice

ground state entropy of the potts antiferromagnet on cyclic strip graphs we present exact calculations of the zero temperature partition function chromatic polynomial and the exponent of the ground state entropy s0 for the q state potts antiferromagnet on families of cyclic and twisted cyclic mobius strip graphs composed of p sided polygons our results suggest a general rule concerning the maximal region in the complex q plane to which one can analytically continue from the physical interval where s0 0 the chromatic zeros and their accumulation set exhibit the rather unusual property of including support for re q 0 and provide further evidence for a relevant conjecture

data driven hierarchical structure kernel for multiscale part based object recognition detecting generic object categories in images and videos are a fundamental issue in computer vision however it faces the challenges from inter and intraclass diversity as well as distortions caused by viewpoints poses deformations and so on to solve object variations this paper constructs a structure kernel and proposes a multiscale part based model incorporating the discriminative power of kernels the structure kernel would measure the resemblance of part based objects in three aspects 1 the global similarity term to measure the resemblance of the global visual appearance of relevant objects 2 the part similarity term to measure the resemblance of the visual appearance of distinctive parts and 3 the spatial similarity term to measure the resemblance of the spatial layout of parts in essence the deformation of parts in the structure kernel is penalized in a multiscale space with respect to horizontal displacement vertical displacement and scale difference part similarities are combined with different weights which are optimized efficiently to maximize the intraclass similarities and minimize the interclass similarities by the normalized stochastic gradient ascent algorithm in addition the parameters of the structure kernel are learned during the training process with regard to the distribution of the data in a more discriminative way with flexible part sizes on scale and displacement it can be more robust to the intraclass variations poses and viewpoints theoretical analysis and experimental evaluations demonstrate that the proposed multiscale part based representation model with structure kernel exhibits accurate and robust performance and outperforms state of the art object classification approaches

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

t 0 partition functions for potts antiferromagnets on square lattice strips with twisted periodic boundary conditions we present exact calculations of the zero temperature partition function for the q state potts antiferromagnet or equivalently the chromatic polynomial for two families of arbitrarily long strip graphs of the square lattice with periodic boundary conditions in the transverse direction and i periodic and ii twisted periodic boundary conditions in the longitudinal direction so that the strip graphs are embedded on i a torus and ii a klein bottle in the limit of infinite length we calculate the exponent of the entropy w q show it to be the same for i and ii and determine its analytic structure

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

spanning trees on the sierpinski gasket we present the numbers of spanning trees on the sierpinski gasket sg d n at stage n with dimension d equal to two three and four the general expression for the number of spanning trees on sg d n with arbitrary d is conjectured the numbers of spanning trees on the generalized sierpinski gasket sg d b n with d 2 and b 3 4 are also obtained

optimal decay rates on the asymptotics of orthogonal polynomial expansions for functions of limited regularities in this paper new and optimal asymptotics on the decay of the coefficients for functions of limited regularity expanded in terms of jacobi and gegenbauer polynomial series are presented for a class of functions with interior singularities the decay of the coefficient is of the same asymptotic order for arbitrary alpha beta 1 which confirms that the decay of the coefficients in the jacobi polynomial series without normalization is a factor of sqrt n slower compared with the chebyshev expansion while for functions with boundary singularities the decay depends on alpha and beta with alpha beta 1 for gegenbauer expansion it is related to the parameter lambda whatever f with interior or boundary singularities all of these asymptotic analysis are optimal moreover under the optimal asymptotic analysis it derives that the truncated spectral expansions with some specific parameters can achieve the optimal convergence rates i e the same as the best polynomial approximation in the sense of absolute maximum error norm numerical examples illustrate the perfect coincidence with the estimates

exact potts model partition functions on ladder graphs abstract we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex ladder graphs i e strips of the square lattice with width ly 2 and arbitrary length lx with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of these ladder graphs and the thermodynamics is discussed by comparison with strip graphs of other widths we analyze how the singularities at the zero temperature critical point of the ferromagnet on infinite length finite width strips depend on the width we point out and study the following noncommutativity at certain special values q s lim n lim q q s z 1 n lim q q s lim n z 1 n it is shown that the potts antiferromagnet on both the infinite length line and ladder graphs with cyclic or mobius boundary conditions exhibits a phase transition at finite temperature if 0 b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros certain properties of the complex temperature phase diagrams are shown to exhibit close connections with those of the model on the square lattice showing that exact solutions on infinite length strips provide a way of gaining insight into these complex temperature phase diagrams

julia sets and complex singularities in hierarchical ising models we study the analytical continuation in the complex plane of free energy of the ising model on diamond like hierarchical lattices it is known 12 13 that the singularities of free energy of this model lie on the julia set of some rational endomorphismf related to the action of the migdal kadanoff renorm group we study the asymptotics of free energy when temperature goes along hyperbolic geodesics to the boundary of an attractive basin off we prove that for almost all with respect to the harmonic measure geodesics the complex critical exponent is common and compute it

the pyramid match kernel efficient learning with sets of features in numerous domains it is useful to represent a single example by the set of the local features or parts that comprise it however this representation poses a challenge to many conventional machine learning techniques since sets may vary in cardinality and elements lack a meaningful ordering kernel methods can learn complex functions but a kernel over unordered set inputs must somehow solve for correspondences generally a computationally expensive task that becomes impractical for large set sizes we present a new fast kernel function called the pyramid match that measures partial match similarity in time linear in the number of features the pyramid match maps unordered feature sets to multi resolution histograms and computes a weighted histogram intersection in order to find implicit correspondences based on the finest resolution histogram cell where a matched pair first appears we show the pyramid match yields a mercer kernel and we prove bounds on its error relative to the optimal partial matching cost we demonstrate our algorithm on both classification and regression tasks including object recognition 3 d human pose inference and time of publication estimation for documents and we show that the proposed method is accurate and significantly more efficient than current approaches

potts models on hierarchical lattices and renormalization group dynamics ii examples and numerical results we obtain the exact renormalization map and plots of lee yang and fisher zeros distributions for potts models on a number of hierarchical lattices the diamond hierarchical lattice a lattice we call spider web the sierpinski gasket and cylinders such models are only examples among those we can study in the general framework of hierarchical lattices developed in a previous paper

chromatic polynomials of large triangular lattices evaluating the q colourings of a lattice is equivalent to solving the q state zero temperature antiferromagnetic potts model this has recently been done exactly for an infinite triangular lattice with q real here the results are extended to the full complex q plane giving the limiting distribution of the zeros of the chromatic polynomial the results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the beraha numbers is noted

exact potts model partition function on strips of the triangular lattice abstract in this paper we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex strip graphs of width ly 2 and arbitrary length of the triangular lattice with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed considering the full generalization to arbitrary complex q and temperature we determine the singular locus b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros comparisons are made with our previous exact calculation of potts model partition functions for the corresponding strips of the square lattice our present calculations yield as special cases several quantities of graph theoretic interest

q plane zeros of the potts partition function on diamond hierarchical graphs we report exact results concerning the zeros of the partition function of the potts model in the complex q plane as a function of a temperature like boltzmann variable v for the m th iterate graphs dm of the diamond hierarchical lattice including the limit m in this limit we denote the continuous accumulation locus of zeros in the q planes at fixed v v0 as bq v0 we apply theorems from complex dynamics to establish the properties of bq v0 for v 1 the zero temperature potts antiferromagnet or equivalently chromatic polynomial we prove that bq 1 crosses the real q axis at i a minimal point q 0 ii a maximal point q 3 iii q 32 27 iv a cubic root that we give with the value q q1 1 638 896 9 and v an infinite number of points smaller than q1 converging to 32 27 from above similar results hold for bq v0 for any 1 0 potts ferromagnet we also provide the computer generated plots of bq v0 at various values of v0 in both the antiferromagnetic and ferromagnetic regimes and compare them to the numerically computed zeros of z d4 q v0

continuity of julia set and its hausdorff dimension of yang lee zeros abstract we show the continuity of the julia set and its hausdorff dimension about a family of rational maps concerning 2 dimensional diamond hierarchical potts models about anti ferromagnetic coupling in statistical mechanics

on connectivity of julia sets of yang lee zeros we prove that the julia sets of the yang lee zeros of the potts model on the diamond hierarchical lattice are connected sets on the complex plane

ground state degeneracy of potts antiferromagnets on two dimensional lattices approach using infinite cyclic strip graphs the q state potts antiferromagnet on a lattice lambda exhibits nonzero ground state entropy s0 kb ln w for sufficiently large q and hence is an exception to the third law of thermodynamics an outstanding challenge has been the calculation of w sq q on the square sq lattice we present here an exact calculation of w on an infinite length cyclic strip of the square lattice which embodies the expected analytic properties of w sq q similar results are given for the kagome lattice

t 0 partition functions for potts antiferromagnets on lattice strips with fully periodic boundary conditions abstract we present exact calculations of the zero temperature partition function for the q state potts antiferromagnet equivalently the chromatic polynomial for families of arbitrarily long strip graphs of the square and triangular lattices with width ly 4 and boundary conditions that are doubly periodic or doubly periodic with reversed orientation i e of torus or klein bottle type these boundary conditions have the advantage of removing edge effects in the limit of infinite length we calculate the exponent of the entropy w q and determine the continuous locus b where it is singular we also give results for toroidal strips involving crossing subgraphs these make possible a unified treatment of torus and klein bottle boundary conditions and enable us to prove that for a given strip the locus b is the same for these boundary conditions

group sensitive multiple kernel learning for object recognition in this paper a group sensitive multiple kernel learning gs mkl method is proposed for object recognition to accommodate the intraclass diversity and the interclass correlation by introducing the group between the object category and individual images as an intermediate representation gs mkl attempts to learn group sensitive multikernel combinations together with the associated classifier for each object category the image corpus from the same category is partitioned into groups images with similar appearance are partitioned into the same group which corresponds to the subcategory of the object category accordingly intraclass diversity can be represented by the set of groups from the same category but with diverse appearances interclass correlation can be represented by the correlation between groups from different categories gs mkl provides a tractable solution to adapt multikernel combination to local data distribution and to seek a tradeoff between capturing the diversity and keeping the invariance for each object category different from the simple hybrid grouping strategy that solves sample grouping and gs mkl training independently two sample grouping strategies are proposed to integrate sample grouping and gs mkl training the first one is a looping hybrid grouping method where a global kernel clustering method and gs mkl interact with each other by sharing group sensitive multikernel combination the second one is a dynamic divisive grouping method where a hierarchical kernel based grouping process interacts with gs mkl experimental results show that performance of gs mkl does not significantly vary with different grouping strategies but the looping hybrid grouping method produces slightly better results on four challenging data sets our proposed method has achieved encouraging performance comparable to the state of the art and outperformed several existing mkl methods

phase transitions on fractals i quasi linear lattices magnetic spin models and resistor networks are studied on certain self similar fractal lattices which are described as quasi linear because they share a significant property of the line finite portions can be isolated from the rest by removal of two points sites in all cases there is no long range order at finite temperature the transition at zero temperature has a discontinuity in the magnetisation and the associated magnetic exponent is equal to the fractal dimensionality d when the lattice reduces to a non branching curve the thermal exponent v 1 y is equal to d when the lattice is a branching curve y is related respectively to the dimensionality of the single channel segments of the curve for the ising model or to the exponent describing the resistivity for models with continuous spin symmetry