odd cycles embedding on folded hypercubes with conditional faulty edges let ff e be the set of ff e 4 and n is even under this assumption we show that every fault free edge of fq n lies on a fault free cycle of every odd length from n 1 to 2 n 1 in terms of the number of tolerant faulty edges and embedding odd cycles in fq n our result improves not only the result in xu and ma 2006 where ff e 0 but also the previous best result gotten by xu et al 2006 where ff e

generalized point interactions in one dimensional quantum mechanics there is a four parameter family of point interactions in one dimensional quantum mechanics they represent all possible self adjoint extensions of the kinetic energy operator if time reversal invariance is imposed the number of parameters is reduced to three one of these point interactions is the familiar function potential but the other generalized ones do not seem to be widely known we present a pedestrian approach to this subject and comment on a recent controversy in the literature concerning the so called interaction we emphasize that there is little resemblance between the interaction and what its name suggests

edge fault tolerant panconnectivity and edge pancyclicity of the complete graph the complete graphs are an important class of graphs and are also fundamental interconnection networks recently fu investigated their edge fault tolerant hamiltonicity and ho et al investigated their edge fault tolerant hamiltonian connectivity in this paper we improve the result of fu and point out that the proof of the result of ho et al fails then we consider the edge fault tolerant panconnectivity of the complete graphs and obtain the following result let f be any set of at most 2n 10 faulty edges in the complete graph k n with n vertices such that every vertex of the graph g k n f is incident with at least three edges and g k 8 e k 4 k 1 0 e k 5 then g is nearly panconnected i e for any two vertices u and v there exists a path connecting u and v in g of any length from 3 to n 1 as a corollary every edge in the graph g lies on a cycle of any length from 4 to n moreover the number 2n 10 of faulty edges tolerated is sharp

fault tolerant cycles embedding in hypercubes with faulty edges let q n be an n dimensional hypercube with f e 5 in this paper we consider the faulty hypercube under the following two additional conditions 1 each vertex is incident to at least two fault free edges and 2 every 4 cycle does not have any pair of non adjacent vertices whose degrees are both two after removing the faulty edges we prove that there exists a fault free cycle of every even length from 4 to 2 n in q n our result improves the result by liu and wang 2014 in terms of the lengths of embedding cycles where under the same conditions a fault free hamiltonian cycle was constructed

embedding hamiltonian cycles in alternating group graphs under conditional fault model in this paper assuming that each node is incident with two or more fault free links we show that an n dimensional alternating group graph can tolerate up to 4n 13 link faults where n 4 while retaining a fault free hamiltonian cycle the proof is computer assisted the result is optimal with respect to the number of link faults tolerated previously without the assumption at most 2n 6 link faults can be tolerated for the same problem and the same graph

fault tolerant edge bipancyclicity of faulty hypercubes under the conditional fault model it is well known that the n dimensional hypercube qn is one of the most versatile and efficient interconnection network architecture yet discovered for building massively parallel or distributed systems let f be the faulty set of qn and let fv fe be the numbers of faulty vertices and faulty edges in f respectively an edge e x y is said to be free if e x y are not in f and a cycle is said to be fault free if there is no faulty vertex or faulty edge on the cycle in this paper we prove that each free edge x y in qn for n 3 lies on a fault free cycle of any even length from 6 to 2 n 2 f v if f v f e 2 n 5 f e n 2 and both x and y are incident to at least two free edges this result confirms a conjecture reported in the literature

embedding paths and cycles in 3 ary n cubes with faulty nodes and links the k ary n cube denoted by q n k is one of the most important interconnection networks for parallel computing in this paper we consider the problem of embedding cycles and paths into faulty 3 ary n cubes let f be a set of faulty nodes and or edges and n 2 we show that when f 2n 2 there exists a cycle of any length from 3 to v q n 3 f in q n 3 f we also prove that when f 2n 3 there exists a path of any length from 2n 1 to v q n 3 f 1 between two arbitrary nodes in q n 3 f since the k ary n cube is regular of degree 2n the fault tolerant degrees 2n 2 and 2n 3 are optimal

conditional edge fault tolerant edge bipancyclicity of hypercubes the hypercube has been one of the most popular interconnection networks for parallel computer communication systems in this paper we assume that each node is incident with at least two fault free links under this assumption we show that every fault free edge lies on a fault free cycle of every even length from 6 to 2 n inclusive even if it has up to 2n 5 link faults the result is optimal with respect to the number of link faults tolerated

fault free hamiltonian cycles in crossed cubes with conditional link faults the crossed cube which is a variation of the hypercube possesses some properties superior to the hypercube in this paper assuming that each node is incident with at least two fault free links we show that an n dimensional crossed cube contains a fault free hamiltonian cycle even if there are up to 2n 5 link faults the result is optimal with respect to the number of link faults tolerated we also verify that the assumption is practically meaningful by evaluating its occurrence probability which is very close to 1

the 2 path bipanconnectivity of hypercubes in this paper we introduce the concept of the k path bi panconnectivity of bipartite graphs it is a generalization of the bi panconnectivity and of the paired many to many k disjoint path cover the 2 path bipanconnectivity with only one exception of the n cube q n n 4 is proved precisely the following result is obtained in an n cube with n 4 given any four vertices u 1 v 1 u 2 v 2 such that two of them are in one partite set and the another two are in the another partite set let s t 5 if c u 1u 2v 1v 2 is a cycle of length 4 and s d u 1 v 1 1 and t d u 2 v 2 1 otherwise where d u v denotes the distance between two vertices u and v and let i and j be any two integers such that both i s 0 and j t 0 are even with i j 2 n then there exist two vertex disjoint u 1 v 1 path p and u 2 v 2 path r with v p i and v r j as consequences many properties of hypercubes such as bipanconnectivity bipanpositionable bipanconnectivity 18 bipancycle connectivity 12 two internally disjoint paths with two given lengths and the 2 disjoint path cover with a path of a given length 21 follow from our result

on embedding cycles into faulty twisted cubes the twisted cube tq n is an alternative to the popular hypercube network recently some interesting properties of tq n were investigated in this paper we study the pancycle problem on faulty twisted cubes let f e and f v be the numbers of faulty edges and faulty vertices in tq n respectively we show that with f e f v 3

long paths and cycles in hypercubes with faulty vertices a fault free path in the n dimensional hypercube q n with f faulty vertices is said to be long if it has length at least 2 n 2f 2 similarly a fault free cycle in q n is long if it has length at least 2 n 2f if all faulty vertices are from the same bipartite class of q n such length is the best possible we show that for every set of at most 2n 4 faulty vertices in q n and every two fault free vertices u and v satisfying a simple necessary condition on neighbors of u and v there exists a long fault free path between u and v this number of faulty vertices is tight and improves the previously known results furthermore we show for every set of at most n 2 10 n 2 1 faulty vertices in q n where n 15 that q n has a long fault free cycle this is a first quadratic bound which is known to be asymptotically optimal

conditional fault tolerant edge bipancyclicity of hypercubes with faulty vertices and edges let f be a faulty set in an n dimensional hypercube q n such that in q n f each vertex is incident to at least two edges and let f v f e be the numbers of faulty vertices and faulty edges in f respectively in this paper we consider the fault tolerant edge bipancyclicity of hypercubes it is shown that each edge in q n f for n 3 lies on a fault free cycle of any even length from 6 to 2 n 2 f v if f v f e 2 n 5 this gives an answer for a problem proposed by yang et al 2016 33 consider hypercubes with mixed faulty edges and faulty vertices under conditional fault model improve the previous results about the fault tolerant edge bipancyclicity of hypercubes give an affirmative answer for a problem proposed by yang et al 2016

edge fault tolerant bipanconnectivity of hypercubes this paper shows that for any two distinct vertices u and v with distance d in the hypercube q n n 3 with at most 2n 5 faulty edges and each vertex incident with least two fault free edges there exist fault free uv paths of length in q n for every with d 4 2 n 1 and d 0 mod2 this result improves some known results on edge fault bipanconnectivity of hypercubes the proof is based on the recursive structure of q n

long paths in hypercubes with conditional node faults let f be a set of f 2 n 5 faulty nodes in an n cube q n such that every node of q n still has at least two fault free neighbors then we show that q n f contains a path of length at least 2 n 2 f 1 respectively 2 n 2 f 2 between any two nodes of odd respectively even distance since the n cube is bipartite the path of length 2 n 2 f 1 or 2 n 2 f 2 turns out to be the longest if all faulty nodes belong to the same partite set as a contribution our study improves upon the previous result presented by j s fu longest fault free paths in hypercubes with vertex faults information sciences 176 2006 759 771 where only n 2 faulty nodes are considered

wigner functions with boundaries we consider the general wigner function for a particle confined to a finite interval and subject to dirichlet boundary conditions we derive the boundary corrections to the stargenvalue equation and to the time evolution equation these corrections can be cast in the form of a boundary potential contributing to the total hamiltonian which together with a subsidiary boundary condition is responsible for the discretization of the energy levels we show that a completely analogous formulation in terms of boundary potentials is also possible in standard operator quantum mechanics and that the wigner and the operator formulations are also in one to one correspondence in the confined case in particular we extend baker s converse construction to bounded systems finally we elaborate on the applications of the formalism to the subject of wigner trajectories namely in the context of collision processes and quantum systems displaying chaotic behavior in the classical limit

edge fault tolerant pancyclicity of arrangement graphs the arrangement graph a n k is a well known interconnection network day and tripathi proved that a n k is pancyclic for n k 2 in this paper we improve this result and we demonstrate that a n k is also pancyclic even if it has no more than k n k 2 faulty edges for n k 2 our result is optimal concerning the edge fault tolerance

constrained steepest descent in the 2 wasserstein metric we study several constrained variational problems in the 2 wasserstein metric for which the set of probability densities satisfying the constraint is not closed for example given a probability density f0 on d and a time step h 0 we seek to minimize i f hs f w 2 2 f0 f over all of the probability densities f that have the same mean and variance as f0 where s f i sthe entropy of f w eprove existence of minimizers we also analyze the induced geometry of the set of densities satisfying the constraint on the variance and means and we determine all of the geodesics on it from this we determine a criterion for convexity of functionals in the induced geometry it turns out for example that the entropy is uniformly strictly convex on the constrained manifold though not uniformly convex without the constraint the problems solved here arose in a study of a variational approach to constructing and studying solutions of the nonlinear kinetic fokker planck equation which is briefly described here and fully developed in a companion paper

pancyclicity and bipancyclicity of conditional faulty folded hypercubes a graph is said to be pancyclic if it contains cycles of every length from its girth to its order inclusive and a bipartite graph is said to be bipancyclic if it contains cycles of every even length from its girth to its order the pancyclicity or the bipancyclicity of a given network is an important factor in determining whether the network s topology can simulate rings of various lengths an n dimensional folded hypercube fq n is an attractive variant of an n dimensional hypercube q n that is obtained by establishing some extra edges between the vertices of q n fq n for any odd n is known to be bipartite in this paper we explore the pancyclicity and bipancyclicity of fq n for any fq n n 2 with at most 2n 3 faulty edges where each vertex is incident to at least two fault free edges we prove that there exists a fault free cycle of every even length from 4 to 2 n and when n 2 is even we prove there also exists a fault free cycle of every odd length from n 1 to 2 n 1 the result is optimal with respect to the number of faulty edges tolerated

on robin boundary conditions and the morse potential in quantum mechanics the physical origin of robin boundary conditions is investigated for wavefunctions at an infinite reflecting wall we consider both schrodinger and phase space quantum mechanics also known as deformation quantization for this simple example of a contact interaction a non relativistic particle moving freely on the half line is treated as moving on the full line in the presence of an infinite potential wall realized as a limit of a morse potential we show that the wavefunctions for the morse states can become those for a free particle on the half line with robin boundary conditions however dirichlet boundary conditions standard walls are obtained unless a mass dependent fine tuning to a reflection resonance is imposed this phenomenon was already observed for piece wise flat potentials so it is not removed by smoothing we argue that it explains why standard quantum walls are standard next we consider the wigner functions the symbols of both diagonal and off diagonal density operator elements of phase space quantum mechanics taking the fine tuned limit we show that our wigner functions do reduce to the expected ones on the half line this confirms that the wigner transform should indeed be unmodified for this contact interaction

the bipancycle connectivity of the hypercube paths and cycles are popular interconnection networks due to their simplicity and low degrees and therefore an efficient way of embedding them into an interconnection network is of particular importance many path embedding aspects and cycle embedding aspects such as hamiltonicity hamiltonian connectivity panconnectivity bipanconnectivity pancyclicity vertex pancyclicity and edge pancyclicity have been proposed and extensively studied in fact these discussions are meaningful for interconnection networks of resource allocated systems or heterogeneous computing systems in this paper we propose a new cycle embedding aspect called bipancycle connectivity that combines the concepts of vertex pancyclicity and bipanconnectivity a bipartite graph is bipancycle connected if an arbitrary pair of vertices x y is contained by common cycles called bipanconnected cycles that include a cycle of every even length ranging from minc x y to n where minc x y denotes the length of the minimum cycle that contains x and y and n is the number of vertices in this paper we introduce a new graph called the cycle of ladders we show that the hypercube is bipancycle connected by presenting algorithms to embed the cycle of ladders into the hypercube

explicitly solvable cases of one dimensional quantum chaos we identify a set of quantum graphs with unique and precisely defined spectral properties called regular quantum graphs although chaotic in their classical limit with positive topological entropy regular quantum graphs are explicitly solvable the proof is constructive we present exact convergent periodic orbit expansions for individual energy levels thus obtaining an analytical solution for the spectrum of regular quantum graphs that is complete explicit and exact

fault tolerant edge pancyclicity of locally twisted cubes the n dimensional locally twisted cube ltq n is a new variant of the hypercube which possesses some properties superior to the hypercube this paper investigates the fault tolerant edge pancyclicity of ltq n and shows that if ltq n n 3 contains at most n 3 faulty vertices and or edges then for any fault free edge e and any integer with 6 2 n f v there is a fault free cycle of length containing the edge e where f v is the number of faulty vertices the result is optimal in some senses the proof is based on the recursive structure of ltq n

on embedding subclasses of height balanced trees in hypercubes a height balanced tree is a rooted binary tree t in which for every vertex v v t the heights of the subtrees rooted at the left and right child of v differ by at most one this difference is called the balance factor of v these trees are extensively used as data structures for sorting and searching we embed several subclasses of height balanced trees of height h in q h 1 under certain conditions in particular if a tree t is such that the balance factor of every vertex in the first three levels is arbitrary 0 or 1 and the balance factor of every other vertex is zero then we prove that t is embeddable in its optimal hypercube with dilation 1 or 2 according to whether it is balanced or not

regular connected bipancyclic spanning subgraphs of hypercubes an n dimensional hypercube q n is a hamiltonian graph in other words q n n 2 contains a spanning subgraph which is 2 regular and 2 connected in this paper we explore yet another strong property of hypercubes we prove that for any integer k with email protected email protected n q n n 3 contains a spanning subgraph which is k regular k connected and bipancyclic we also obtain the result that every mesh p mxp n m n 2 is bipancyclic which is used to prove the property above

many to many disjoint paths in faulty hypercubes this paper considers the problem of many to many disjoint paths in the hypercube q n with f v faulty vertices and f e faulty edges and obtains the following result for any integer k with 1 k n 1 any two sets s and t of k fault free vertices in different parts if f v f e n k 1 then there exist k disjoint fault free s t paths in q n which contains at least 2 n 2 f v vertices this result is optimal in the worst case

solving for the wigner functions of the morse potential in deformation quantization we consider the time independent wigner functions of phase space quantum mechanics also known as deformation quantization for a morse potential first we find them by solving the eigenvalue equations using a method that can be applied to potentials that are polynomial in an exponential a mellin transform converts the eigenvalue equations to difference equations and factorized solutions are found directly for all values of the parameters the symbols found this way are of both diagonal and off diagonal density operator elements in the energy basis the wigner transforms of the density matrices built from the known wavefunctions are then shown to confirm the solutions

a systematic approach for embedding of hamiltonian cycles through a prescribed edge in locally twisted cubes the locally twisted cube interconnection network has been recognized as an attractive alternative to the hypercube network previously the locally twisted cube has been shown to contain a hamiltonian cycle the main contribution of this paper is to provide the necessary and sufficient conditions for determining a characterization of permutations of link dimensions constructing hamiltonian cycles in a locally twisted cube for those permutations we propose a linear algorithm for finding a hamiltonian cycle through a given edge as a result we obtain a lower bound for the number of hamiltonian cycles through a given edge in an n dimensional locally twisted cube

linear array and ring embeddings in conditional faulty hypercubes the n dimensional hypercube qn is a graph having 2n vertices labeled from 0 to 2n 1 two vertices are connected by an edge if their binary labels differ in exactly one bit position in this paper we consider the faulty hypercube qn with n 3 that each vertex of qn is incident to at least two nonfaulty edges based on this requirement we prove that qn contains a hamiltonian path joining any two different colored vertices even if it has up to 2n 5 edge faults moreover we show that there exists a path of length 2n 2 between any two the same colored vertices in this faulty qn furthermore we also prove that the faulty qn still contains a cycle of every even length from 4 to 2n inclusive

hamiltonian cycles in hypercubes with faulty edges let f be a set of faulty edges in hypercube q n with f 5 we prove that there still exists a fault free hamiltonian cycle in q n if the following two conditions are satisfied 1 the degree of every vertex is at least two and 2 there do not exist a pair of nonadjacent vertices in a 4 cycle whose degrees are both two after faulty edges are removed

embedding hamiltonian paths in k ary n cubes with conditional edge faults it is well known that the k ary n cube has been one of the most efficient interconnection networks for distributed memory parallel systems a k ary n cube is bipartite if and only if k is even in this paper we consider the faulty k ary n cube with even k 4 and n 2 such that each vertex of the k ary n cube is incident with at least two healthy edges based on this requirement we prove that the k ary n cube contains a hamiltonian path joining every pair of vertices which are in different parts even if it has up to 4n 6 edge faults

conditional matching preclusion sets abstract the matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings for many interconnection networks the optimal sets are precisely those induced by a single vertex in this paper we look for obstruction sets beyond these sets we introduce the conditional matching preclusion number of a graph it is the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost perfect matchings we find this number and classify all optimal sets for several basic classes of graphs