classifications of exact structures and cohen macaulay finite algebras abstract we give a classification of all exact structures on a given idempotent complete additive category using this we investigate the structure of an exact category with finitely many indecomposables we show that the relation of the grothendieck group of such a category is generated by ar conflations moreover we obtain an explicit classification of 1 gorenstein projective finite iwanaga gorenstein algebras 2 cohen macaulay finite orders and more generally 3 cotilting modules u with u of finite type in the appendix we develop the ar theory of exact categories over a noetherian complete local ring and relate the existence of ar conflations to the ar duality and dualizing varieties

heat kernel coefficients of the laplace operator on the d dimensional ball we present a very quick and powerful method for the calculation of heat kernel coefficients it makes use of rather common ideas as integral representations of the spectral sum mellin transforms non trivial commutation of series and integrals and skillful analytic continuation of zeta functions on the complex plane we apply our method to the case of the heat kernel expansion of the laplace operator on a d dimensional ball with either dirichlet neumann or in general robin boundary conditions the final formulas are quite simple using this case as an example we illustrate in detail our scheme which serves for the calculation of an in principle arbitrary number of heat kernel coefficients in any situation when the basis functions are known we provide a complete list of new results for the coefficients b3 b10 corresponding to the d dimensional ball with all the mentioned boundary conditions and d 3 4 5

eigenvalues and degeneracies for n dimensional tensor spherical harmonics symmetric transverse traceless tensor harmonics of arbitrary rank are constructed on spheres sn of dimensionality n 3 and the associated eigenvalues of the laplacian are computed it is shown that these tensor harmonics span the space of symmetric transverse traceless tensors on sn and are eigenfunctions of the quadratic casimir operator of the group o n 1 the dimensionalities of the eigenspaces of the laplacian are computed for harmonics of rank 1 and rank 2

singular equivalences of functor categories via auslander buchweitz approximations abstract the aim of this paper is to construct singular equivalences between functor categories as a special case we show that there exists a singular equivalence arising from a cotilting module t namely the singularity category of t t and that of mod t are triangle equivalent in particular the canonical module over a commutative noetherian ring r induces a singular equivalence between cm r and mod r which generalizes matsui takahashi s theorem our result is based on a sufficient condition for an additive category a and its subcategory x so that the canonical inclusion x a induces a singular equivalence d sg a d sg x which is a functor category version of xiao wu chen s theorem

law of large numbers for the possibilistic mean value abstract a law of large numbers for the possibilistic mean value of a variable in a possibility space is presented an example shows that the convergence in distribution under a definition involving the possibilistic mean value of the sample average to a variable with a certain distribution cannot be replaced in general by convergence either almost surely or in necessity even so the usual presentation of the law of large numbers as a statement that holds in necessity follows from this result

a simple recurrence for the higher derivatives of the hurwitz zeta function a recurrent formula which allows the calculation of the asymptotic series expansion of any derivative m z a m z a zm of the hurwitz zeta function z a is obtained in particular the first terms of the series corresponding to n a in inverse powers of a are explicitly given for n 0 1 2 3 knowledge of these expressions is fundamental e g in the zeta function regularization procedure

dirac operator on a disk with global boundary conditions we compute the functional determinant for a dirac operator in the presence of an abelian gauge field on a bidimensional disk under global boundary conditions of the type introduced by atiyah patodi singer we also discuss the connection between our result and the index theorem

antisymmetric tensor fields on spheres functional determinants and non local counterterms the hodge de rham laplacian on spheres acting on antisymmetric tensor fields is considered explicit expressions for the spectrum are derived in a quite direct way confirming previous results associated functional determinants and the heat kernel expansion are evaluated using this method new non local counterterms in the quantum effective action are obtained which can be expressed in terms of betti numbers

on singular equivalences of morita type abstract stable equivalences of morita type preserve many interesting properties and are proved to be the appropriate concept for studying equivalences between stable categories recently the singularity category attained much attraction and xiao wu chen and long gang sun gave an appropriate definition of singular equivalence of morita type we shall show that under some conditions singular equivalences of morita type have some biadjoint functor properties and preserve positive degree hochschild homology

explicit vector spherical harmonics on the 3 sphere we build a family of explicit one forms on s3 which are shown to form a new complete set of eigenmodes for the laplace de rham operator

arithmetical degeneracies in simple quantum systems the authors examine the accidental level degeneracies occurring in the quantum mechanical problem of a free particle moving in a polyhedral box when the problem is integrable some remarkable properties of the distribution of degeneracies are studied in several two three and four dimensional examples and are related to well known problems of number theory the numerical results of exact enumerations are compared with analytical predictions or with conjectured expressions in some cases they consider in particular the asymptotic scaling form of the degeneracy distribution up to some maximal energy e and the maximal degeneracy occurring for energies less than some given e

on the zeta function regularization of a two dimensional series of epstein hurwitz type as a further step in the general program of zeta function regularization of multiseries expressions some original formulas are provided for the analytic continuation to any value of s of two dimensional series of epstein hurwitz type namely n1 n2 0 a1 n1 c1 2 a2 n2 c2 2 s where the aj are positive reals and the cj are not simultaneously nonpositive integers they come out from a generalization to hurwitz functions of the zeta function regularization theorem of the author and romeo phys rev d 40 436 1989 for ordinary zeta functions for s k 0 2 with k 1 2 3 the final results are in fact expressed in terms of hurwitz zeta functions only for general s they also involve bessel functions a partial numerical investigation of the different terms of the exact algebraic equations is also carried out as a by product the series n 0exp a n c 2 a c 0 is conveniently calculated in terms of them

homological theory of idempotent ideals let be an artin algebra u and a two sided ideal of then u is the trace of a projective module p in we study how the homological properties of the categories of finitely generated modules over the three rings u and the endomorphism ring of p are related we give some applications of the ideas developed in the paper to the study of quasi hereditary algebras

vector fields on a disk with mixed boundary conditions vector fields on a disk satisfying two types of boundary conditions are studied these boundary conditions are selected by brst invariance in electrodynamics they also appear in the de rham complex the construction of the harmonic expansion is presented the eigenfunctions of the vector laplace operator are expressed in terms of fields satisfying pure dirichlet or robin boundary conditions for the case of four dimensional disk several first coefficients of the heat kernel expansion are computed an error in the analytical expression by branson and gilkey is corrected

the law of large numbers for fuzzy processes and the estimation problem abstract we study how the law of large numbers can be extended to the case of fuzzy numbers we work with l fuzzy sets where l is a cl commutative monoid with an operation we define the independence of fuzzy numbers we find when is the operation of the lattice that the arithmetical mean of independent fuzzy numbers does not converge we think that this case could be a limit one we show for particular cases of the operation that generally convergence holds we build estimators for fuzzy models we particularly consider the minimum fuzziness estimator we study the convergence of these estimators and compare them by way of the fuzzy subsets they induce we lastly study the case where we have dependent fuzzy numbers and particularly the case of linearly filtered autoindependent fuzzy processes

spectral action for torsion with and without boundaries we derive a commutative spectral triple and study the spectral action for a rather general geometric setting which includes the skew symmetric torsion and the chiral bag conditions on the boundary the spectral action splits into bulk and boundary parts in the bulk we clarify certain issues of the previous calculations show that many terms in fact cancel out and demonstrate that this cancellation is a result of the chiral symmetry of spectral action on the boundary we calculate several leading terms in the expansion of spectral action in four dimensions for vanishing chiral parameter of the boundary conditions and show that 0 is a critical point of the action in any dimension and at all orders of the expansion

strong law of large numbers for t normed arithmetics we prove a strong law of large numbers for sequences of pairwise i i d fuzzy random variables addition is given by the extension principle associated with a general continuous triangular norm our result includes the slln for levelwise sums obtained when the chosen triangular norm is the minimum

localization of triangulated categories and derived categories the notion of quotient and localization of abelian categories by dense subcategories i e serre classes was introduced by gabriel and plays an important role in ring theory 6 131 the notion of triangulated categories was introduced by grothendieck and developed by verdier 9 16 and is recently useful in representation theory 8 4 143 the quotient of triangulated categories by epaisse subcategories is constructed in 16 both quotients were indicated by grothendieck and they resemble each other in this paper we will consider triangulated categories and derived categories from the point of view of localization of abelian categories verdier gave a condition which is equivalent to the one that a quotient functor has a right adjoint and considered a relation between tpaisse subcategories 16 we show that localization of triangulated categories is similarly defined and have a relation between localizations and cpaisse subcategories beilinson bernstein and deligne introduced the notion of t structure which is similar to torsion theory in abelian categories 2 we in particular consider a stable t structure which is an epaisse subcategory and deal with a correspondence between localizations of triangulated categories and stable r structures and then recollement in the sense of 2 is equivalent to bilocalization next we show that quotient and localization of abelian categories induce quotient and localization of its derived categories in section 1 we recall standard notations and terminologies of quotient and localization of abelian categories in section 2 we define localization of triangulated categories and consider a relation between localizations and stable r structures theorem 2 6 in section 3 we show that if a a c is a quotient of abelian categories then d a d a c is a quotient of triangulated categories where or b theorem 3 2 moreover

relative singularity categories i auslander resolutions abstract let r be an isolated gorenstein singularity with a non commutative resolution a end r r m in this paper we show that the relative singularity category r a of a has a number of pleasant properties such as being hom finite moreover it determines the classical singularity category d s g r of buchweitz and orlov as a certain canonical quotient category if r has finite cm type which includes for example kleinian singularities then we show the much more surprising result that d s g r determines r aus r where aus r is the corresponding auslander algebra the proofs of these results use dg algebras a koszul duality and the new concept of dg auslander algebras which may be of independent interest

a note on zadeh s extensions let f rn rn and f f rn f r n be zadeh s extension of f to the space of fuzzy compact sets f rn the aim of this paper is to show that if f is continuous then f f rn d f rn d is also continuous d being the supremum over hausdorff distances between their corresponding level sets

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

zeta function for the laplace operator acting on forms in a ball with gauge boundary conditions the laplace operator acting on antisymmetric tensor fields in a d dimensional euclidean ball is studied gauge invariant local boundary conditions absolute and relative ones in the language of gilkey are considered the eigenfuctions of the operator are found explicitly for all values of d using in a row a number of basic techniques as mellin transforms deformation and shifting of the complex integration contour and pole compensation the zeta function of the operator is obtained from its expression in particular 0 and 0 are evaluated exactly a table is given in the paper for d 3 4 8 the functional determinants and casimir energies are obtained for d 3 4 6

a law of large numbers for fuzzy numbers abstract we study the problem if 1 2 are fuzzy numbers with modal values m1 m2 then what is the strongest t norm for which where mn m1 mn n the arithmetic mean 1 n n i is defined via sup t norm convolution and nes denotes necessity

on possibilistic mean value and variance of fuzzy numbers dubois and prade introduced the mean value of a fuzzy number as a closed interval bounded by the expectations calculated from its upper and lower distribution functions in this paper introducing the notations of lower possibilistic and upper possibilistic mean values we definine the interval valued possibilistic mean and investigate its relationship to the interval valued probabilistic mean we also introduce the notation of crisp possibilistic mean value and crisp possibilistic variance of continuous possibility distributions which are consistent with the extension principle we also show that the variance of linear combination of fuzzy numbers can be computed in a similar manner as in probability theory

zeta function regularization for a scalar field in a compact domain we express the zeta function associated with the laplacian operator on s1r m in terms of the zeta function associated with the laplacian on m where m is a compact connected riemannian manifold this gives formulae for the partition function of the associated physical model at low and high temperature for any compact domain m furthermore we provide an exact formula for the zeta function at any value of r when m is a d dimensional box or a d dimensional torus this allows a rigorous calculation of the zeta invariants and the analysis of the main thermodynamic functions associated with the physical models at finite temperature

singularity categories and singular equivalences for resolving subcategories let mathcal x be a resolving subcategory of an abelian category in this paper we investigate the singularity category mathsf d sg underline mathcal x mathsf d b mathsf mod underline mathcal x mathsf k b mathsf proj mathsf mod underline mathcal x of the stable category underline mathcal x of mathcal x we consider when the singularity category is triangle equivalent to the stable category of gorenstein projective objects and when the stable categories of two resolving subcategories have triangle equivalent singularity categories applying this to the module category of a gorenstein ring we prove that the complete intersections over which the stable categories of resolving subcategories have trivial singularity categories are the simple hypersurface singularities of type mathsf a 1 we also generalize several results of yoshino on totally reflexive modules

analysis of an inhomogeneous generalized epstein hurwitz zeta function with physical applications the inhomogeneous generalized epstein hurwitz like multidimensional series em s a1 am c1 cm c2 n1 nm 0 a1 n1 c1 2 am nm cm 2 c2 s is investigated by means of a nontrivial asymptotic recurrence it is reduced to the one dimensional case f s a b2 n 0 n a 2 b2 s which is then studied in full detail in particular asymptotic expansions for f and its derivatives f s and f a together with analytical continuations of the same in the variable s are explicitly obtained using zeta function techniques several plots and tables of the numerical results are given some explicit applications to the regularization by means of hurwitz zeta functions of different problems that have appeared recently in the physical literature are considered

conformal invariance and the regularised one loop effective action a unique family gn 1 of conformal invariants polynomial in the extrinsic curvature of a hypersurface embedded in an n dimensional riemannian manifold m is constructed in the quantum theory of a conformally coupled scalar field on a manifold m with boundary delta m the counterterm that regularises the one loop effective action contains members of gn 1 integrated over delta m the counterterms through n 4 are given explicitly and a derivation of the n 4 boundary contribution is given based on a flat space result

an extension of the chowla selberg formula useful in quantizing with the wheeler de witt equation the two dimensional inhomogeneous zeta function series with homogeneous part of the most general epstein type sigma m n in z am2 bmn cn2 q s is analytically continued in the variable s using zeta function techniques a simple formula is obtained which extends the chowla selberg formula to inhomogeneous epstein zeta functions the new expression is then applied to solve the problem of computing the determinant of the basic differential operator that appears in an attempt at quantizing gravity by using the wheeler de witt equation in 2 1 dimensional spacetime with the torus topology

zeta function determinant of the laplace operator on the d dimensional ball we present a direct approach for the calculation of functional determinants of the laplace operator on balls dirichlet and robin boundary conditions are considered using this approach formulas for any value of the dimension d of the ball can be obtained quite easily explicit results are presented here for dimensionsd 2 3 4 5 and 6

zeta function regularization is uniquely defined and well hawking s zeta function regularization procedure 1977 is shown to be rigorously and uniquely defined thus putting an end to the spreading lore about different difficulties associated with it basic misconceptions misunderstandings and errors which keep appearing in important scientific journals when dealing with this beautiful regularization method and other analytical procedures are clarified and corrected

multiple harmonic oscillator zeta functions the partition function for a quantum mechanical system can at least formally be expressed as a high temperature series in terms of the zeta function of the system s hamiltonian this is done explicitly for a system of non interacting harmonic oscillators the hamilton zeta function belongs to a class of zeta functions sigma f ni n1 omega 1 n2 omega 2 nn omega n a s summed over non negative integers ni very little is known about these zeta functions and an investigation of some of their properties is presented

on functional determinants of laplacians in polygons and simplicial complexes the functional determinant of an elliptic operator with positive discrete spectrum may be defined ase z 0 wherez s the zeta function is the sum sum limits n lambda n delta analytically continued ins in this paperz 0 is calculated for the laplace operator with dirichlet boundary conditions inside polygons with the topology of a disc in the euclidean plane our results are complementary to earlier investigations of the determinants on smooth surfaces with smooth boundaries our expression can be viewed as the energy for a system of static point particles corresponding to the corners of the polygon with self energy and pair interaction energy we have completely explicit closed expressions for triangles and regular polygons with an arbitrary number of sides among these there are five special cases three triangles the square and the circled where thez 0 are known by other means one special case fixes an integration constant and the other provide four independent analytical checks on our calculation

complete determination of the singularity structure of zeta functions series of extended epstein type provide examples of non trivial zeta functions with important physical applications the regular part of their analytic continuation is seen to be a convergent or an asymptotic series their singularity structure is completely determined in terms of the wodzicki residue in its generalized form which is proven to yield the residua of all the poles of the zeta function and not just that of the rightmost pole obtainable from the dixmier trace the calculation is a very down to earth application of these powerful functional analytical methods in physics

the functional integral for fields in a cavity the functional integral for a scalar field confined in a cavity and subjected to linear boundary conditions is discussed herein it is shown how the functional measure can be conveniently dealt with by modifying the classical action with boundary corrections the nonuniqueness of the boundary actions is described with a three parameter family of them giving identical boundary conditions in some cases the corresponding green s function will define a kind of generalized gaussian measure on function space the vacuum energy is discussed paying due attention to its anomalous scale dependence and the physical issues involved are considered

a law of large numbers for fuzzy numbers in a banach space abstract we study a law of large numbers for mutually t related fuzzy number in a banach space where t is an archimedean t norm and extend earlier results of fuller and triesch in this area

strong ellipticity and spectral properties of chiral bag boundary conditions we prove strong ellipticity of chiral bag boundary conditions on even dimensional manifolds from a knowledge of the heat kernel in an infinite cylinder some basic properties of the zeta function are analysed on cylindrical product manifolds of arbitrary even dimension

symmetric tensor eigenspectrum of the laplacian on n spheres the eigenvalues and degeneracies of the covariant laplacian acting on symmetric tensors of rank m 2 defined on n spheres with n 3 are given

zeta function continuation and the casimir energy on odd and even dimensional spheres the zeta function continuation method is applied to compute the casimir energy on spheressn both odd and even dimensional spheres are studied for the appropriate conformally modified laplacian the casimir energy is shown to be finite for all dimensions even though generically it should diverge in odd dimensions due to the presence of a pole in the associated zeta function s the residue of this pole is computed and shown to vanish in our case an explicit analytic continuation of s is constructed for all values ofn this enables us to calculate exactly and we find that the casimir energy vanishes in all even dimensions for odd dimensions is never zero but alternates in sign asn increases through odd values some results are also derived for the casimir energy of other operators of laplacian type