a note on the optimal quadriphase sequences families in this note by using a modification of the families b and c we obtain a larger family of optimal quadriphase sequences d over z4 in contrast to the families b and c the family d has the same length and the same maximal nontrival correlation value but with double the size

low firing rates in a compartmental model neuron a nonlinear relationship is obtained between the membrane potential of a compartmental model neuron as a function of input stimulation this is used to show how shunting effects can lead to low output firing rates in associative memory networks

on the inequivalence of generalized preparata codes if m is odd and sigma in aut gf 2 m is such that x rightarrow x sigma 2 1 is 1 1 there is a 2 m 1 1 2 m l 2m 2 nonlinear binary code p sigma having minimum distance 5 all the codes p sigma have the same distance and weight enumerators as the usual preparata codes which rise as p sigma when x sigma x 2 it is shown that p sigma and p tau are equivalent if and only if tau sigma pm 1 and aut p sigma is determined

z4 kerdock codes orthogonal spreads and extremal euclidean line sets when m is odd spreads in an orthogonal vector space of type omega 2m 2 2 are related to binary kerdock codes and extremal line sets in rr 2 m 1 with prescribed angles spreads in a 2m dimensional binary symplectic vector space are related to kerdock codes over zz 4 and extremal line sets in cc 2 m with prescribed angles these connections involve binary real and complex geometry associated with extraspecial 2 groups a geometric map from symplectic to orthogonal spreads is shown to induce the gray map from a corresponding zz 4 kerdock code to its binary image these geometric considerations lead to the construction for any odd composite m of large numbers of zz 4 kerdock codes they also produce new zz 4 linear kerdock and preparata codes 1991 mathematics subject classification primary 94b60 secondary 51m15 20c99

new families of binary sequences with low correlation for a positive integer n new families s and u of binary sequences of period 2 sup n 1 with low correlations are proposed where for some positive integer e s is defined for odd n e and u for even n e the family s has four valued correlations and is a generalization of the family of gold like sequences introduced by s boztas and p v kumar see ibid vol 40 p 532 7 1994 the family u which is also a generalization of the sequence family defined by p udaya polyphase and frequency hopping sequences obtained from finite rings ph d dissertation dept elec eng indian inst technol kanpur 1992 has six valued correlations the relationship between gold like sequences and gold sequences is the same as the relationship between the family s and the family constructed from the binary sequences partially contributed by r gold see ibid vol it 14 p 154 6 1968 t kasami see coordinated sci lab univ of illinois urbana champaign tech rep r 285 ad 632574 1966 and welch using a lifting idea no j s and kumar p v ibid vol 35 p 371 9 1989 for the families s and u families of binary sequences with the same correlation distributions and large linear span are also constructed

fuzzy nets and their application fuzzy net theory is enriched by the introduction of the fuzzy net of fuzzy sets the limsup liminf and limit of a fuzzy net of fuzzy sets are defined and their various properties are discussed alternative characterizations based on the notion of a fuzzy net are provided for several fuzzy topological concepts including open and closed fuzzy sets fuzzy continuity maps with closed fuzzy graph open fuzzy mapping etc thus the net theoretic approach is shown to be a promising tool in fuzzy topology

on the construction of binary sequence families with low correlation and large sizes in this paper we revisit a method to produce binary sequences using the most significant bit map from z4 to the binary field this method is useful for the construction of binary sequences with low correlation and large family size there may be more cases where starting with z4 could help researchers design new low correlation sequences for code division multiple access application

new constructions of quadratic bent functions in polynomial form new quadratic bent functions in polynomial form are constructed in this paper the constructions give new boolean bent generalized boolean bent and p ary bent functions based on z 4 valued quadratic forms a simple method provides several new constructions of generalized boolean bent functions from these generalized boolean bent functions a method is presented to transform them into boolean bent and semi bent functions moreover many new p ary bent functions can also be obtained by applying similar methods

bbz _4 valued quadratic forms and quaternary sequence families in this paper zopf4 valued quadratic forms defined on a vector space over gf 2 are studied a classification of such forms is established distinguishing zopf4 valued quadratic forms only by their rank and whether the associated bilinear form is alternating this result is used to compute the distribution of certain exponential sums which occur frequently in the analysis of quaternary codes and quaternary sequence sets the concept is applied as follows when t 0 or m is odd the correlation distribution of family s t consisting of quaternary sequences of length 2 m 1 is established then motivated by practical considerations a subset s t of family s t is defined and the correlation distribution of family s t is given for odd and even m

a new approach to fuzzy topological spaces and various fuzzy topological notions defined by nets n fuzzy spaces introduced by the authors are further investigated various fuzzy topological motions such as fuzzy local compactness functions with closed fuzzy graph fuzzy subcontinuity inverse fuzzy subcontinuity etc are studied in the light of n i these notions are found to be more coherent in n i than in the chang s category

several classes of codes and sequences derived from a bbz_ 4 valued quadratic form let m and k be positive integers with m gcd m k being odd for a r and b l the exponential sum x litr ax 2bx2k 1 is studied systematically in this paper where i 1 r gr 4 m is a galois ring l is the teichmuller set of r and tr is the trace function from the galois ring r to z4 through the discussions on the solutions of certain equations and the newly developed theory of z4 valued quadratic forms the distribution of the exponential sum is completely determined as its applications we can determine the lee weight and hamming weight distributions of a class of codes ck over z4 and the correlation distribution of a quaternary sequence family uk respectively furthermore the hamming weight distributions of the binary codes obtained from ck under the most significant bit msb and gray maps are also determined for the msb map sequences of uk the nontrivial maximal correlation value is given and the correlation distribution is determined for the gray map sequences of uk it should be noted that the distribution of the exponential sum for the case gcd m k 1 is obtained for the first time and then the corresponding codes and sequences are novel

t convergence of nets of l fuzzy sets and its applications abstract in this paper the theory of convergence on nets of l fuzzy sets which is a generalization of the convergence theory of fuzzy nets introduced by sarma and ajmal is established some important and interesting results in l fuzzy topology are obtained by means of the theory

binary sequences with gold like correlation but larger linear span a new construction of optimal binary sequences identical to the well known family of gold sequences in terms of maximum nontrivial correlation magnitude and family size but having larger linear span is presented the distribution of correlation values is determined for every odd integer spl tau spl ges 3 the construction provides a family that contains 2 sup spl tau 1 cyclically distinct sequences each of period 2 sup spl tau 1 the maximum nontrivial correlation magnitude equals 2 spl tau 1 sup 2 1 with one exception each of the sequences in the family has linear span at least spl tau sup 2 spl tau 2 compared to 2 spl tau for gold sequences the sequences are easily implemented using a quaternary shift register followed by a simple feedforward nonlinearity

optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings we construct new sequences over finite rings having optimal hamming correlation properties these sequences are useful in frequency hopping multiple access fhma spread spectrum communication systems our constructions can be classified into linear and nonlinear categories both giving optimal hamming correlations according to lempel greenberger 1974 bound the nonlinear sequences have large linear complexity and can be seen as a generalized version of gmw sequences over fields

further results on the optimal sequence family mathcal ip _ 8 over 8 ary q pam constellation a family mathcal ip 8 of sequences over the 8 ary quadrature pulse amplitude modulation q pam constellation with asymptotically optimal correlation property was presented by anand and kumar in 2008 this is the only known family of sequences over the quadrature amplitude modulation qam constellation whose correlation magnitude asymptotically achieves the welch bound in this paper a larger family mathcal ip 8 text new of sequences with the same correlation magnitude as that of mathcal ip 8 but double family size is obtained moreover the correlation distributions of mathcal ip 8 and mathcal ip 8 text new are also determined for two particular cases in terms of the property of a class of exponential sums over galois rings it is also shown that more classes of qam and q pam sequences with low correlation and larger family size can be obtained from our approach

new applications of l nets in l fuzzy multifunctions in this paper we will give new applications of l net in l fuzzy multifunctions using l net we will further study l fuzzy continuous almost continuous and r continuous multifunctions some interesting characterizations are obtained

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

the correlation distribution of quaternary sequences of period 2 2 n 1 family a is a family of sequences of period 2n 1 over zi the ring of integers modulo 4 this family has optimal correlation properties and its correlation distribution is well known two related families of quaternary sequences are the families b and c these are families of sequences over z4 of period 2 2n 1 in recent years new families of quaternary sequences of period 2 2n 1 have been constructed by modifying the sequence families b and c in a nonlinear way this has resulted in a new family d of sequences of period 2 2n 1 which has optimal correlation properties but until now the correlation distribution of this family has not been known in this paper we completely determine the correlation distribution of family d by making use of properties of exponential sums

4 phase sequences with near optimum correlation properties two families of four phase sequences are constructed using irreducible polynomials over z sub 4 family a has period l 2 sup r 1 size l 2 and maximum nontrivial correlation magnitude c sub max

fuzzy almost continuity and its pointwise characterization by dual points and fuzzy nets abstract we introduce the notions of fuzzy almost continuity and fuzzy weak continuity at a fuzzy point following the pointwise approach we investigate their fuzzy topological properties we also obtain a pointwise characterization of fuzzy almost continuous functions by dual points and fuzzy nets the main idea behind this work is to establish a local theory in the study of continuity and its weaker forms in general fuzzy topological spaces

net convergence in fuzzy neighborhood spaces our aim is to introduce a concept of net convergence in the well behaved subcategory of lowen fuzzy neighborhood spaces of the category of fuzzy topological spaces in doing so we notice that our present notion of net convergence in addition to katsaras and papageorgeou net convergence is generalized sequential convergence we also apply this notion of net convergence in fuzzy neighborhood groups and fuzzy neighborhood linear spaces

generic construction of binary sequences of period 2n with optimal odd correlation magnitude based on quaternary sequences of odd period n binary sequences with low odd correlation have important applications in communication systems to reduce interference in this paper using the interleaving technique we present a generic connection between binary sequences with low odd correlation and quaternary sequences with low even correlation as a result some new binary sequences with optimal odd auto correlation magnitude are obtained besides two sets consisting of 2 n 1 binary sequences of period 2 2 n 1 with the maximum odd correlation magnitude 2 n 1 2 2 are derived which are the first two optimal classes of binary sequence sets achieving the sarwate bound on the odd correlation magnitude in the literature

fuzzy sets and functions on fuzzy spaces abstract we generally follow the terminology of azad j math anal appl 82 1981 14 32 and ming j math anal appl 76 1980 571 599 in addition to the fundamental concepts for fuzzy sets we emphasize the usefulness of the concepts of fuzzy point fuzzy elementhood for fuzzy sets a and b a new characterization is given for relations a b and a b this knowledge permits us to combine the two definitions of fuzzy point fuzzy elementhood in the third section some results are given concerning various special types of fuzzy sets in fuzzy topological spaces and the fuzzy semi regular fuzzy regular spaces defined by azad in the last section the definitions of h almost continuous w almost open functions urysohn space which are defined by hussain wilansky and noiri respectively are extended to fuzzy sets furthermore some results are obtained in the functions of the fuzzy topological spaces defined by azad and those are defined here

on the correlation distributions of the optimal quaternary sequence family cal u and the optimal binary sequence family cal v recently new optimal families s and u of quaternary sequences have been presented and the optimal binary sequence family v obtained from family s under gray map has been investigated as well the two sequence families u and v are optimal with respect to the well known sidelnikov bound and welch bound but their exact correlation distributions are not known until now in this paper their exact correlation distributions are completely determined in some cases by making use of exponential sums and the theory of z4 valued quadratic forms

on fuzzy topological spaces abstract in this paper a few separation properties and some aspects of subspace fuzzy topology have been studied where both the crisp and the fuzzy elements have been taken into consideration since the conventional definition of compactness is not quite meaningful in hausdorff fuzzy spaces as introduced by us a new more natural definition of proper compactness is given and a few properties resulting from this are established

two new families of optimal binary sequences obtained from quaternary sequences in this paper we present two optimal binary families of sequences of length 2n 1 and 2 2n 1 for odd integer n they are obtained as the images of proposed optimal quaternary sequences under the most significant bit and the gray maps the first family has 2n 1 sequences of length 2n 1 and the identical correlation distribution to that of gold sequences and gold like sequences and the second family of sequences of length 2 2n 1 has 2n sequences and the same correlation values as those of kerdock sequences

fuzzy subcontinuity inverse fuzzy subcontinuity and a new category of fuzzy topological spaces the notions of subcontinuity and inverse subcontinuity are extended to the fuzzy setting adopting fuzzy net theoretic approach various properties of fuzzy subcontinuous and inversely fuzzy subcontinuous mappings are studied a category of fuzzy topological spaces is constructed canonical example is constructed by using fuzzy translation of fuzzy numbers introduced by rodabaugh to show that fuzzy subcontinuity does not imply kfuzzy continuity in the process n compactness is characterized and fuzzy unit interval is shown to be n compact also a direct proof is provided showing that fuzzy addition of fuzzy numbers is fuzzy continuous

various fuzzy topological notions and fuzzy almost continuity abstract we extend the notion of almost continuity in the sense of husain 1966 to the fuzzy setting we obtain a pointwise characterization of the notion by dual point and fuzzy net also we formulate conditions under which the two notions of fuzzy almost continuity the one defined in this paper and the one defined by ajmal and azad 1990 turn out to be equivalent the main idea behind this work is to show that using the localized approach developed in earlier papers the study of fuzzy almost continuity can be carried out as successfully as that of its counterpart in general topology in the process we exhibit the compatibility of various existing fuzzy topological notions such as fuzzy regularity c i connectedness i 3 4 c i local connectedness i 3 4 fuzzy t 1 axiom fuzzy t 2 axiom etc

some important applications of nets of l fuzzy sets in this paper some compactness adherence point and accumulation point in l fuzzy topological space are characterized by means of the nets of l fuzzy subsets

a new approach to fuzzy topological spaces and fuzzy perfect mappings abstract a net theoretic approach is provided for defining a fuzzy topological space fuzzy topological spaces so defined form a category the main features and advantages of this category which is denoted by n here are discussed the notion of a perfect mapping is introduced and studied in the fuzzy setting it is found that in n a fts is n compact iff any mapping x x where x is any singleton is fuzzy perfect a fuzzy continuous mapping x y from an n compact fts x to a hausdorff fts y is fuzzy perfect in n however these results do not hold in c the chang s category in general it is found that the divergences in the study of fuzzy perfect mappings disappear if the study is carried out in n instead of c

new optimal quadriphase sequences with larger linear span in this paper two new optimal families s and u of quadriphase sequences are presented compared to the family a constructed by boztas and the family d investigated by tang respectively the proposed families have the same optimal correlation properties and family size but larger linear spans

open maps closed maps and local compactness in fuzzy topological spaces on donne des caracterisations des applications floues ouvertes et fermees et on montre que la normalite definie par hutton est invariante sous des applications f fermees f continues

bethe ansatz equations for quantum chains combining different representations of su 3 the general expression for the local matrix of a quantum chain with the site space in any representation of is obtained this is made by generalizing from the fundamental representation and imposing the fulfilment of the yang baxter equation with these operators and using a generalization of the nested bethe ansatz the bethe equations for a multistate quantum chain combining two arbitrary representations of su 3 are obtained

generic construction of quaternary sequences of period 2n with low correlation from quaternary sequences of odd period n in this paper a simple but generic method is proposed for transforming any family of quaternary sequences with low correlation of any odd period n to another family of quaternary sequences of period 2n with low correlation as an application of the generic method to sequence family a a new optimal quaternary sequence family with length 2 2n 1 family size 2n 1 and maximal nontrivial correlation value 2 n 1 2 2 where n is an odd integer is obtained most notably unlike all the known optimal quaternary sequence families the new family has a unique property that the odd integers 1 3 and the even integers 0 2 are allocated alternatively in all the sequences

on fuzzy convergence classes abstract the concept of fuzzy point and its quasi coincidence neighborhood have been introduced in 1 using these notions we give the theory of fuzzy convergence classes especially a characterization of fuzzy topology via the fuzzy convergence classes and relative analysis are given

category n and functions with closed fuzzy graph abstract a net theoretic approach for defining a fuzzy topological space is provided the fuzzy topological spaces so obtained form a category which we denote by n in this paper it is found that the limitations exhibited by the chang category particularly with respect to projections closure of a fuzzy set compactness etc no longer exist in n the notion of closed fuzzy graph is investigated the divergences shown by this notion are immediately put to rest if the study is carried out in n instead of c the chang category

an exponential number of generalized kerdock codes if n 1 is an odd composite integer then there are at least 2 1 2 n pairwise inequivalent binary error correcting codes of length 2 n size 2 2 n and minimum distance 2 n 1 2 1 2 n 1

new integrable quantum chains combining different kinds of spins a general procedure to generate new integrable hamiltonians combining to any kind of spins distributed arbitrarily on the line is given as a concrete application anisotropic chains formed by spin 1 2 and spin 1 operators at alternating sites are presented and solved exactly by the bethe ansatz ba the authors compute the ground state and excitation energies and momentum the higher order ba equations are derived depending on the choice these new hamiltonians exhibit or not conformal invariance in their low energy spectrum